Dirichlet series
| L(s) = 1 | − 40·4-s + 112·9-s + 104·11-s + 880·16-s − 220·25-s − 216·29-s − 4.48e3·36-s − 4.16e3·44-s − 1.40e4·64-s − 1.87e3·71-s − 6.42e3·79-s + 4.64e3·81-s + 1.16e4·99-s + 8.80e3·100-s − 3.67e3·109-s + 8.64e3·116-s − 2.61e3·121-s + 127-s + 131-s + 137-s + 139-s + 9.85e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 5·4-s + 4.14·9-s + 2.85·11-s + 55/4·16-s − 1.75·25-s − 1.38·29-s − 20.7·36-s − 14.2·44-s − 27.5·64-s − 3.12·71-s − 9.14·79-s + 6.37·81-s + 11.8·99-s + 44/5·100-s − 3.22·109-s + 6.91·116-s − 1.96·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 57.0·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 5^{20} \cdot 7^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.66434\times 10^{29}\) |
| Root analytic conductor: | \(5.37688\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 5^{20} \cdot 7^{40} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.002208321231\) |
| \(L(\frac12)\) | \(\approx\) | \(0.002208321231\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{10} \) |
| 5 | \( 1 + 44 p T^{2} + 29617 T^{4} + 2460112 T^{6} + 8886278 p^{2} T^{8} + 47007208 p^{4} T^{10} + 8886278 p^{8} T^{12} + 2460112 p^{12} T^{14} + 29617 p^{18} T^{16} + 44 p^{25} T^{18} + p^{30} T^{20} \) | |
| 7 | \( 1 \) | |
| good | 3 | \( ( 1 - 56 T^{2} + 794 p T^{4} - 73778 T^{6} + 1706881 T^{8} - 5126476 p^{2} T^{10} + 1706881 p^{6} T^{12} - 73778 p^{12} T^{14} + 794 p^{19} T^{16} - 56 p^{24} T^{18} + p^{30} T^{20} )^{2} \) |
| 11 | \( ( 1 - 26 T + 2344 T^{2} - 39260 T^{3} - 44437 T^{4} - 11813492 T^{5} - 44437 p^{3} T^{6} - 39260 p^{6} T^{7} + 2344 p^{9} T^{8} - 26 p^{12} T^{9} + p^{15} T^{10} )^{4} \) | |
| 13 | \( ( 1 - 11382 T^{2} + 69082830 T^{4} - 287052185808 T^{6} + 900043216534725 T^{8} - 2216936700366038516 T^{10} + 900043216534725 p^{6} T^{12} - 287052185808 p^{12} T^{14} + 69082830 p^{18} T^{16} - 11382 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 17 | \( ( 1 - 18318 T^{2} + 174928646 T^{4} - 1304812398728 T^{6} + 8438722092560269 T^{8} - 45734748022547850996 T^{10} + 8438722092560269 p^{6} T^{12} - 1304812398728 p^{12} T^{14} + 174928646 p^{18} T^{16} - 18318 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 19 | \( ( 1 + 43166 T^{2} + 940837125 T^{4} + 712492946488 p T^{6} + 141145266089486066 T^{8} + \)\(11\!\cdots\!20\)\( T^{10} + 141145266089486066 p^{6} T^{12} + 712492946488 p^{13} T^{14} + 940837125 p^{18} T^{16} + 43166 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 23 | \( ( 1 - 71290 T^{2} + 106241607 p T^{4} - 53647271029088 T^{6} + 869997711169400142 T^{8} - \)\(11\!\cdots\!84\)\( T^{10} + 869997711169400142 p^{6} T^{12} - 53647271029088 p^{12} T^{14} + 106241607 p^{19} T^{16} - 71290 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 54 T + 85314 T^{2} + 5010492 T^{3} + 3590822385 T^{4} + 168821978284 T^{5} + 3590822385 p^{3} T^{6} + 5010492 p^{6} T^{7} + 85314 p^{9} T^{8} + 54 p^{12} T^{9} + p^{15} T^{10} )^{4} \) | |
| 31 | \( ( 1 + 161458 T^{2} + 14078008401 T^{4} + 824619099347712 T^{6} + 35924144663503995438 T^{8} + \)\(12\!\cdots\!68\)\( T^{10} + 35924144663503995438 p^{6} T^{12} + 824619099347712 p^{12} T^{14} + 14078008401 p^{18} T^{16} + 161458 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 113898 T^{2} + 12485339513 T^{4} - 848687834784128 T^{6} + 53768342114840304494 T^{8} - \)\(27\!\cdots\!72\)\( T^{10} + 53768342114840304494 p^{6} T^{12} - 848687834784128 p^{12} T^{14} + 12485339513 p^{18} T^{16} - 113898 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 396400 T^{2} + 84845234933 T^{4} + 12094542419183040 T^{6} + \)\(12\!\cdots\!50\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!50\)\( p^{6} T^{12} + 12094542419183040 p^{12} T^{14} + 84845234933 p^{18} T^{16} + 396400 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 306350 T^{2} + 50916207093 T^{4} - 5322617174838632 T^{6} + \)\(43\!\cdots\!82\)\( T^{8} - \)\(32\!\cdots\!60\)\( T^{10} + \)\(43\!\cdots\!82\)\( p^{6} T^{12} - 5322617174838632 p^{12} T^{14} + 50916207093 p^{18} T^{16} - 306350 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 47 | \( ( 1 - 353120 T^{2} + 81269257654 T^{4} - 13564790125665194 T^{6} + \)\(18\!\cdots\!93\)\( T^{8} - \)\(20\!\cdots\!08\)\( T^{10} + \)\(18\!\cdots\!93\)\( p^{6} T^{12} - 13564790125665194 p^{12} T^{14} + 81269257654 p^{18} T^{16} - 353120 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 612454 T^{2} + 251712421305 T^{4} - 69244774609214848 T^{6} + \)\(14\!\cdots\!66\)\( T^{8} - \)\(24\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!66\)\( p^{6} T^{12} - 69244774609214848 p^{12} T^{14} + 251712421305 p^{18} T^{16} - 612454 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 59 | \( ( 1 + 1214074 T^{2} + 730047369721 T^{4} + 292328358982556800 T^{6} + \)\(87\!\cdots\!74\)\( T^{8} + \)\(20\!\cdots\!12\)\( T^{10} + \)\(87\!\cdots\!74\)\( p^{6} T^{12} + 292328358982556800 p^{12} T^{14} + 730047369721 p^{18} T^{16} + 1214074 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 333980 T^{2} + 234140938049 T^{4} + 54538754775301072 T^{6} + \)\(22\!\cdots\!66\)\( T^{8} + \)\(38\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!66\)\( p^{6} T^{12} + 54538754775301072 p^{12} T^{14} + 234140938049 p^{18} T^{16} + 333980 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 67 | \( ( 1 - 3218 T^{2} - 48896888727 T^{4} - 33181943235248608 T^{6} + \)\(93\!\cdots\!06\)\( T^{8} + \)\(20\!\cdots\!92\)\( T^{10} + \)\(93\!\cdots\!06\)\( p^{6} T^{12} - 33181943235248608 p^{12} T^{14} - 48896888727 p^{18} T^{16} - 3218 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 71 | \( ( 1 + 468 T + 625863 T^{2} + 32609488 T^{3} + 10386329358 T^{4} - 77558080051528 T^{5} + 10386329358 p^{3} T^{6} + 32609488 p^{6} T^{7} + 625863 p^{9} T^{8} + 468 p^{12} T^{9} + p^{15} T^{10} )^{4} \) | |
| 73 | \( ( 1 - 2238084 T^{2} + 2529024940153 T^{4} - 1918853291003376528 T^{6} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(47\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!26\)\( p^{6} T^{12} - 1918853291003376528 p^{12} T^{14} + 2529024940153 p^{18} T^{16} - 2238084 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 1606 T + 2388468 T^{2} + 2275264208 T^{3} + 2100969503759 T^{4} + 1507373343439412 T^{5} + 2100969503759 p^{3} T^{6} + 2275264208 p^{6} T^{7} + 2388468 p^{9} T^{8} + 1606 p^{12} T^{9} + p^{15} T^{10} )^{4} \) | |
| 83 | \( ( 1 - 3630638 T^{2} + 6381511863877 T^{4} - 86811290431823096 p T^{6} + \)\(59\!\cdots\!06\)\( T^{8} - \)\(37\!\cdots\!52\)\( T^{10} + \)\(59\!\cdots\!06\)\( p^{6} T^{12} - 86811290431823096 p^{13} T^{14} + 6381511863877 p^{18} T^{16} - 3630638 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 89 | \( ( 1 + 4189668 T^{2} + 9192796269977 T^{4} + 13381265983504366096 T^{6} + \)\(14\!\cdots\!90\)\( T^{8} + \)\(11\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!90\)\( p^{6} T^{12} + 13381265983504366096 p^{12} T^{14} + 9192796269977 p^{18} T^{16} + 4189668 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 8346158 T^{2} + 31893652323686 T^{4} - 73615692743331346152 T^{6} + \)\(11\!\cdots\!69\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{10} + \)\(11\!\cdots\!69\)\( p^{6} T^{12} - 73615692743331346152 p^{12} T^{14} + 31893652323686 p^{18} T^{16} - 8346158 p^{24} T^{18} + p^{30} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.95165236190713779567781901435, −1.82904049002007289691857292996, −1.76729813172289085050901146135, −1.69388957067861105106622343801, −1.65560336166641093541759156126, −1.52295975891162072822723161577, −1.49433800779627892859452900938, −1.38586121818835185465814144605, −1.38089289537605744773914122072, −1.19328236535738638054366556766, −1.17004862600813045799085994247, −1.14982087010032639628013698355, −1.14355927410256486757505670828, −1.09186662904159029448950018037, −1.08985987295734178183529712152, −1.05424298120639048162498049394, −0.987732365968233515949791167575, −0.811291039078910323122177489371, −0.50919014212979268343927276855, −0.33604575836454291714440573453, −0.26472880875290460837086603221, −0.12264614034763938718455466265, −0.083784716168923052720160243936, −0.080451029830261702255471840556, −0.04555991268815519150830141633, 0.04555991268815519150830141633, 0.080451029830261702255471840556, 0.083784716168923052720160243936, 0.12264614034763938718455466265, 0.26472880875290460837086603221, 0.33604575836454291714440573453, 0.50919014212979268343927276855, 0.811291039078910323122177489371, 0.987732365968233515949791167575, 1.05424298120639048162498049394, 1.08985987295734178183529712152, 1.09186662904159029448950018037, 1.14355927410256486757505670828, 1.14982087010032639628013698355, 1.17004862600813045799085994247, 1.19328236535738638054366556766, 1.38089289537605744773914122072, 1.38586121818835185465814144605, 1.49433800779627892859452900938, 1.52295975891162072822723161577, 1.65560336166641093541759156126, 1.69388957067861105106622343801, 1.76729813172289085050901146135, 1.82904049002007289691857292996, 1.95165236190713779567781901435
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.