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Properties

Label 2-637-7.4-c1-0-26
Degree 22
Conductor 637637
Sign −0.947+0.318i-0.947 + 0.318i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 − 1.89i)2-s + (0.879 − 1.52i)3-s + (−1.38 + 2.39i)4-s + (1.05 + 1.82i)5-s − 3.83·6-s + 1.67·8-s + (−0.0460 − 0.0797i)9-s + (2.30 − 3.99i)10-s + (2.88 − 4.99i)11-s + (2.43 + 4.21i)12-s − 13-s + 3.71·15-s + (0.939 + 1.62i)16-s + (0.820 − 1.42i)17-s + (−0.100 + 0.174i)18-s + (−1.33 − 2.31i)19-s + ⋯
L(s)  = 1  + (−0.771 − 1.33i)2-s + (0.507 − 0.879i)3-s + (−0.691 + 1.19i)4-s + (0.471 + 0.817i)5-s − 1.56·6-s + 0.591·8-s + (−0.0153 − 0.0265i)9-s + (0.728 − 1.26i)10-s + (0.869 − 1.50i)11-s + (0.702 + 1.21i)12-s − 0.277·13-s + 0.958·15-s + (0.234 + 0.406i)16-s + (0.198 − 0.344i)17-s + (−0.0236 + 0.0410i)18-s + (−0.306 − 0.530i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.947+0.318i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.947+0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.947+0.318i-0.947 + 0.318i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(508,⋅)\chi_{637} (508, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.947+0.318i)(2,\ 637,\ (\ :1/2),\ -0.947 + 0.318i)

Particular Values

L(1)L(1) ≈\approx 0.194151−1.18713i0.194151 - 1.18713i
L(12)L(\frac12) ≈\approx 0.194151−1.18713i0.194151 - 1.18713i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1+T 1 + T
good2 1+(1.09+1.89i)T+(−1+1.73i)T2 1 + (1.09 + 1.89i)T + (-1 + 1.73i)T^{2}
3 1+(−0.879+1.52i)T+(−1.5−2.59i)T2 1 + (-0.879 + 1.52i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.05−1.82i)T+(−2.5+4.33i)T2 1 + (-1.05 - 1.82i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.88+4.99i)T+(−5.5−9.52i)T2 1 + (-2.88 + 4.99i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.820+1.42i)T+(−8.5−14.7i)T2 1 + (-0.820 + 1.42i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.33+2.31i)T+(−9.5+16.4i)T2 1 + (1.33 + 2.31i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.21+5.56i)T+(−11.5+19.9i)T2 1 + (3.21 + 5.56i)T + (-11.5 + 19.9i)T^{2}
29 1+6.04T+29T2 1 + 6.04T + 29T^{2}
31 1+(−2.56+4.43i)T+(−15.5−26.8i)T2 1 + (-2.56 + 4.43i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.87+4.97i)T+(−18.5+32.0i)T2 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2}
41 1−7.14T+41T2 1 - 7.14T + 41T^{2}
43 1+4.47T+43T2 1 + 4.47T + 43T^{2}
47 1+(−5.89−10.2i)T+(−23.5+40.7i)T2 1 + (-5.89 - 10.2i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.72−2.98i)T+(−26.5−45.8i)T2 1 + (1.72 - 2.98i)T + (-26.5 - 45.8i)T^{2}
59 1+(−6.59+11.4i)T+(−29.5−51.0i)T2 1 + (-6.59 + 11.4i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.12−5.40i)T+(−30.5+52.8i)T2 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.87−6.70i)T+(−33.5−58.0i)T2 1 + (3.87 - 6.70i)T + (-33.5 - 58.0i)T^{2}
71 1−13.6T+71T2 1 - 13.6T + 71T^{2}
73 1+(7.75−13.4i)T+(−36.5−63.2i)T2 1 + (7.75 - 13.4i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.561+0.971i)T+(−39.5+68.4i)T2 1 + (0.561 + 0.971i)T + (-39.5 + 68.4i)T^{2}
83 1+4.96T+83T2 1 + 4.96T + 83T^{2}
89 1+(−0.573−0.992i)T+(−44.5+77.0i)T2 1 + (-0.573 - 0.992i)T + (-44.5 + 77.0i)T^{2}
97 1−6.97T+97T2 1 - 6.97T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.32722345106143925351662677287, −9.370077566814129526873652076358, −8.648623607312374155165243981327, −7.85393730178682292883308460848, −6.74384502107590723426557680719, −5.93101955299640339810650154856, −3.98695217612706761310461563794, −2.80787146396598819215575541938, −2.22785786452515907565182451515, −0.853671758562385527432953524487, 1.62522407169955369505322153151, 3.69571411363732470667833999736, 4.73155189098100163473561871852, 5.60589964704065411291038783734, 6.69792323491332796414820081447, 7.51000306532162450512909826561, 8.530418744560861332241066980316, 9.192181146286901120380174530537, 9.723238677073559872716231636619, 10.22940627416415395709724418238

Graph of the ZZ-function along the critical line