Dirichlet series
| L(s) = 1 | − 3·3-s + 5·7-s + 3·9-s + 15·19-s − 15·21-s − 5·25-s + 3·31-s − 4·37-s − 26·43-s + 7·49-s − 45·57-s + 19·61-s + 15·63-s + 5·67-s + 3·73-s + 15·75-s + 17·79-s − 9·93-s − 27·103-s + 19·109-s + 12·111-s + 11·121-s + 127-s + 78·129-s + 131-s + 75·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.88·7-s + 9-s + 3.44·19-s − 3.27·21-s − 25-s + 0.538·31-s − 0.657·37-s − 3.96·43-s + 49-s − 5.96·57-s + 2.43·61-s + 1.88·63-s + 0.610·67-s + 0.351·73-s + 1.73·75-s + 1.91·79-s − 0.933·93-s − 2.66·103-s + 1.81·109-s + 1.13·111-s + 121-s + 0.0887·127-s + 6.86·129-s + 0.0873·131-s + 6.50·133-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{12} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14776\times 10^{8}\) |
| Root analytic conductor: | \(2.16684\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{12} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6039988528\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6039988528\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T + 2 p T^{2} + p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{9} + 2 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) | |
| 7 | \( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 360 T^{5} + 757 T^{6} - 360 p T^{7} + 149 p^{2} T^{8} - 55 p^{3} T^{9} + 18 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} - p^{9} T^{18} + p^{11} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} + p^{9} T^{18} - p^{11} T^{22} + p^{12} T^{24} \) | |
| 13 | \( ( 1 - 5 T + 12 T^{2} + 5 T^{3} - 181 T^{4} + 840 T^{5} - 1847 T^{6} + 840 p T^{7} - 181 p^{2} T^{8} + 5 p^{3} T^{9} + 12 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )( 1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 840 T^{5} - 1847 T^{6} - 840 p T^{7} - 181 p^{2} T^{8} - 5 p^{3} T^{9} + 12 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 17 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} - p^{9} T^{18} + p^{11} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 2520 T^{5} + 4789 T^{6} - 2520 p T^{7} + 809 p^{2} T^{8} - 208 p^{3} T^{9} + 45 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )( 1 - 7 T + 30 T^{2} - 77 T^{3} - 31 T^{4} + 1680 T^{5} - 11171 T^{6} + 1680 p T^{7} - 31 p^{2} T^{8} - 77 p^{3} T^{9} + 30 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 23 | \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} + p^{9} T^{18} - p^{11} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 7 T + 18 T^{2} + 91 T^{3} - 1195 T^{4} + 5544 T^{5} - 1763 T^{6} + 5544 p T^{7} - 1195 p^{2} T^{8} + 91 p^{3} T^{9} + 18 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} )( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} + 4620 T^{5} + 26881 T^{6} + 4620 p T^{7} - 271 p^{2} T^{8} - 184 p^{3} T^{9} - 15 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 37 | \( ( 1 - T + p T^{2} )^{6}( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} - 6930 T^{5} - 79253 T^{6} - 6930 p T^{7} + 269 p^{2} T^{8} + 260 p^{3} T^{9} + 63 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 13 T + 126 T^{2} + 1079 T^{3} + 8609 T^{4} + 65520 T^{5} + 481573 T^{6} + 65520 p T^{7} + 8609 p^{2} T^{8} + 1079 p^{3} T^{9} + 126 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} - p^{9} T^{18} + p^{11} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} + p^{9} T^{18} - p^{11} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} - p^{9} T^{18} + p^{11} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - T + p T^{2} )^{6}( 1 - 13 T + 108 T^{2} - 611 T^{3} + 1355 T^{4} + 19656 T^{5} - 338183 T^{6} + 19656 p T^{7} + 1355 p^{2} T^{8} - 611 p^{3} T^{9} + 108 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 67 | \( ( 1 - 16 T + 189 T^{2} - 1952 T^{3} + 18569 T^{4} - 166320 T^{5} + 1416997 T^{6} - 166320 p T^{7} + 18569 p^{2} T^{8} - 1952 p^{3} T^{9} + 189 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )( 1 + 11 T + 54 T^{2} - 143 T^{3} - 5191 T^{4} - 47520 T^{5} - 174923 T^{6} - 47520 p T^{7} - 5191 p^{2} T^{8} - 143 p^{3} T^{9} + 54 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 71 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 10 T + 27 T^{2} + 460 T^{3} - 6571 T^{4} + 32130 T^{5} + 158383 T^{6} + 32130 p T^{7} - 6571 p^{2} T^{8} + 460 p^{3} T^{9} + 27 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )( 1 + 7 T - 24 T^{2} - 679 T^{3} - 3001 T^{4} + 28560 T^{5} + 418993 T^{6} + 28560 p T^{7} - 3001 p^{2} T^{8} - 679 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 79 | \( ( 1 - 13 T + 90 T^{2} - 143 T^{3} - 5251 T^{4} + 79560 T^{5} - 619451 T^{6} + 79560 p T^{7} - 5251 p^{2} T^{8} - 143 p^{3} T^{9} + 90 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} - 55692 p T^{7} + 2705 p^{2} T^{8} + 568 p^{3} T^{9} - 63 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} ) \) | |
| 83 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{8} T^{16} - p^{9} T^{18} + p^{11} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 131670 p T^{7} - 9211 p^{2} T^{8} - 28 p^{3} T^{9} + 99 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} - 131670 T^{5} - 949913 T^{6} - 131670 p T^{7} - 9211 p^{2} T^{8} + 28 p^{3} T^{9} + 99 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} ) \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.51447582420886923937571185805, −3.37258109901128815261671395288, −3.35294808595322940210372375943, −3.27373279188164110938561534319, −3.22817992535607256970585174642, −3.05862249007377743979105855422, −2.91891955372534257262192106204, −2.79794804723528566991133051525, −2.61481452692486861748329339723, −2.48563377857187289054524889699, −2.35405315537534965170532394021, −2.33448833503271989844988892125, −2.17184770714984148648077317148, −1.99355695149944963531779476616, −1.87877513215208232386808074232, −1.86360197929613531367787620979, −1.67003301764087493956531098424, −1.39888155373144070234945186153, −1.33119893371115696374504825659, −1.23429580222491473633445683700, −1.07684277891283665934026799877, −1.00158621950417210554511101372, −0.64860535172667476546892407768, −0.52805753287209597589104280182, −0.099596926920832666803251385110, 0.099596926920832666803251385110, 0.52805753287209597589104280182, 0.64860535172667476546892407768, 1.00158621950417210554511101372, 1.07684277891283665934026799877, 1.23429580222491473633445683700, 1.33119893371115696374504825659, 1.39888155373144070234945186153, 1.67003301764087493956531098424, 1.86360197929613531367787620979, 1.87877513215208232386808074232, 1.99355695149944963531779476616, 2.17184770714984148648077317148, 2.33448833503271989844988892125, 2.35405315537534965170532394021, 2.48563377857187289054524889699, 2.61481452692486861748329339723, 2.79794804723528566991133051525, 2.91891955372534257262192106204, 3.05862249007377743979105855422, 3.22817992535607256970585174642, 3.27373279188164110938561534319, 3.35294808595322940210372375943, 3.37258109901128815261671395288, 3.51447582420886923937571185805
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.