Properties

Label 32-490e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.104\times 10^{43}$
Sign $1$
Analytic cond. $3.01697\times 10^{9}$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·11-s − 4·16-s + 8·23-s − 12·25-s − 8·37-s − 8·43-s + 56·53-s − 64·67-s + 16·71-s + 6·81-s − 40·107-s − 8·113-s + 236·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 96·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 7.23·11-s − 16-s + 1.66·23-s − 2.39·25-s − 1.31·37-s − 1.21·43-s + 7.69·53-s − 7.81·67-s + 1.89·71-s + 2/3·81-s − 3.86·107-s − 0.752·113-s + 21.4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 7.23·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(3.01697\times 10^{9}\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1165018782\)
\(L(\frac12)\) \(\approx\) \(0.1165018782\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{4} \)
5 \( 1 + 12 T^{2} + 72 T^{4} + 324 T^{6} + 1454 T^{8} + 324 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 \)
good3 \( 1 - 2 p T^{4} + 17 T^{8} + 182 p T^{12} - 572 T^{16} + 182 p^{5} T^{20} + 17 p^{8} T^{24} - 2 p^{13} T^{28} + p^{16} T^{32} \)
11 \( ( 1 - 6 T + 31 T^{2} - 102 T^{3} + 378 T^{4} - 102 p T^{5} + 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
13 \( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 68653746 p^{4} T^{20} + 266977 p^{8} T^{24} + 714 p^{12} T^{28} + p^{16} T^{32} \)
17 \( 1 - 192 T^{4} - 111236 T^{8} + 6807744 T^{12} + 12704251398 T^{16} + 6807744 p^{4} T^{20} - 111236 p^{8} T^{24} - 192 p^{12} T^{28} + p^{16} T^{32} \)
19 \( ( 1 + 90 T^{2} + 4089 T^{4} + 122298 T^{6} + 2679044 T^{8} + 122298 p^{2} T^{10} + 4089 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 4 T + 8 T^{2} - 80 T^{3} - 58 T^{4} + 1396 T^{5} - 1920 T^{6} + 6452 T^{7} + 96123 T^{8} + 6452 p T^{9} - 1920 p^{2} T^{10} + 1396 p^{3} T^{11} - 58 p^{4} T^{12} - 80 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 80906 p^{2} T^{10} + 2685 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 168 T^{2} + 13620 T^{4} - 703224 T^{6} + 25584998 T^{8} - 703224 p^{2} T^{10} + 13620 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 4 T + 8 T^{2} + 136 T^{3} + 2569 T^{4} + 160 p T^{5} + 12376 T^{6} + 210804 T^{7} + 3581856 T^{8} + 210804 p T^{9} + 12376 p^{2} T^{10} + 160 p^{4} T^{11} + 2569 p^{4} T^{12} + 136 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1146544 p^{2} T^{10} + 17802 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 651228 p T^{9} + 55896 p^{2} T^{10} + 20696 p^{3} T^{11} + 3425 p^{4} T^{12} + 32 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 + 2626 T^{4} + 11315169 T^{8} + 37752482306 T^{12} + 67553145100100 T^{16} + 37752482306 p^{4} T^{20} + 11315169 p^{8} T^{24} + 2626 p^{12} T^{28} + p^{16} T^{32} \)
53 \( ( 1 - 28 T + 392 T^{2} - 4584 T^{3} + 50633 T^{4} - 470816 T^{5} + 3841240 T^{6} - 30898524 T^{7} + 237353440 T^{8} - 30898524 p T^{9} + 3841240 p^{2} T^{10} - 470816 p^{3} T^{11} + 50633 p^{4} T^{12} - 4584 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 320 T^{2} + 48156 T^{4} + 4600000 T^{6} + 314762150 T^{8} + 4600000 p^{2} T^{10} + 48156 p^{4} T^{12} + 320 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 378 T^{2} + 67917 T^{4} - 7492902 T^{6} + 552750680 T^{8} - 7492902 p^{2} T^{10} + 67917 p^{4} T^{12} - 378 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 32 T + 512 T^{2} + 6244 T^{3} + 72845 T^{4} + 802048 T^{5} + 7862664 T^{6} + 70013484 T^{7} + 587621764 T^{8} + 70013484 p T^{9} + 7862664 p^{2} T^{10} + 802048 p^{3} T^{11} + 72845 p^{4} T^{12} + 6244 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 4 T + 94 T^{2} + 964 T^{3} - 1158 T^{4} + 964 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( 1 - 22400 T^{4} + 253668732 T^{8} - 1943585257600 T^{12} + 11498706287626118 T^{16} - 1943585257600 p^{4} T^{20} + 253668732 p^{8} T^{24} - 22400 p^{12} T^{28} + p^{16} T^{32} \)
79 \( ( 1 - 344 T^{2} + 61500 T^{4} - 7270888 T^{6} + 650307974 T^{8} - 7270888 p^{2} T^{10} + 61500 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - 2957648820318 p^{4} T^{20} + 318852241 p^{8} T^{24} - 25638 p^{12} T^{28} + p^{16} T^{32} \)
89 \( ( 1 + 522 T^{2} + 127569 T^{4} + 19418370 T^{6} + 2046124772 T^{8} + 19418370 p^{2} T^{10} + 127569 p^{4} T^{12} + 522 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 3868 T^{4} + 90482118 T^{8} + 3868 p^{4} T^{12} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.01494764491534141323671381010, −2.95014803284386186120612938607, −2.76432225100279497959633066689, −2.58728097240323325863422558624, −2.50961151118831201339554291386, −2.49250451162971682008234480879, −2.46249042916843609400826222463, −2.41935403177018642446999542105, −2.36459909135410225334741138048, −2.25067714102503716781817096060, −2.22930726183610971488578736264, −1.94144264173562908136179055762, −1.68933632908054256151371260313, −1.63450734209227672882924386958, −1.51671869273612253336074160188, −1.50275644696088147352417613253, −1.45246998753796732920901584994, −1.34339764918394175443992410601, −1.34152259920313253303084369709, −1.20603263692282532759071027584, −1.01085069019790722678782605985, −0.995629204024991212220061288654, −0.76536856014748442773618711789, −0.35151834417837811716797330220, −0.02562902663432083958228799685, 0.02562902663432083958228799685, 0.35151834417837811716797330220, 0.76536856014748442773618711789, 0.995629204024991212220061288654, 1.01085069019790722678782605985, 1.20603263692282532759071027584, 1.34152259920313253303084369709, 1.34339764918394175443992410601, 1.45246998753796732920901584994, 1.50275644696088147352417613253, 1.51671869273612253336074160188, 1.63450734209227672882924386958, 1.68933632908054256151371260313, 1.94144264173562908136179055762, 2.22930726183610971488578736264, 2.25067714102503716781817096060, 2.36459909135410225334741138048, 2.41935403177018642446999542105, 2.46249042916843609400826222463, 2.49250451162971682008234480879, 2.50961151118831201339554291386, 2.58728097240323325863422558624, 2.76432225100279497959633066689, 2.95014803284386186120612938607, 3.01494764491534141323671381010

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.