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Properties

Label 2-637-91.81-c1-0-6
Degree 22
Conductor 637637
Sign −0.962+0.270i-0.962 + 0.270i
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  + (−1.15 + 1.99i)2-s − 2.16·3-s + (−1.65 − 2.86i)4-s + (1.08 + 1.87i)5-s + (2.49 − 4.32i)6-s + 2.99·8-s + 1.69·9-s − 4.99·10-s + 4.90·11-s + (3.57 + 6.19i)12-s + (1.41 + 3.31i)13-s + (−2.34 − 4.06i)15-s + (−0.151 + 0.262i)16-s + (3.57 + 6.19i)17-s + (−1.95 + 3.38i)18-s − 2.16·19-s + ⋯
L(s)  = 1  + (−0.814 + 1.41i)2-s − 1.25·3-s + (−0.825 − 1.43i)4-s + (0.484 + 0.839i)5-s + (1.01 − 1.76i)6-s + 1.06·8-s + 0.565·9-s − 1.57·10-s + 1.47·11-s + (1.03 + 1.78i)12-s + (0.391 + 0.920i)13-s + (−0.606 − 1.05i)15-s + (−0.0378 + 0.0655i)16-s + (0.868 + 1.50i)17-s + (−0.460 + 0.797i)18-s − 0.497·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.962+0.270i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.962+0.270i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.962+0.270i-0.962 + 0.270i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(263,⋅)\chi_{637} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.962+0.270i)(2,\ 637,\ (\ :1/2),\ -0.962 + 0.270i)

Particular Values

L(1)L(1) ≈\approx 0.0817953−0.594009i0.0817953 - 0.594009i
L(12)L(\frac12) ≈\approx 0.0817953−0.594009i0.0817953 - 0.594009i
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+(−1.41−3.31i)T 1 + (-1.41 - 3.31i)T
good2 1+(1.15−1.99i)T+(−1−1.73i)T2 1 + (1.15 - 1.99i)T + (-1 - 1.73i)T^{2}
3 1+2.16T+3T2 1 + 2.16T + 3T^{2}
5 1+(−1.08−1.87i)T+(−2.5+4.33i)T2 1 + (-1.08 - 1.87i)T + (-2.5 + 4.33i)T^{2}
11 1−4.90T+11T2 1 - 4.90T + 11T^{2}
17 1+(−3.57−6.19i)T+(−8.5+14.7i)T2 1 + (-3.57 - 6.19i)T + (-8.5 + 14.7i)T^{2}
19 1+2.16T+19T2 1 + 2.16T + 19T^{2}
23 1+(−0.302+0.524i)T+(−11.5−19.9i)T2 1 + (-0.302 + 0.524i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.15−1.99i)T+(−14.5+25.1i)T2 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.57−6.19i)T+(−15.5−26.8i)T2 1 + (3.57 - 6.19i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.30−7.45i)T+(−18.5−32.0i)T2 1 + (4.30 - 7.45i)T + (-18.5 - 32.0i)T^{2}
41 1+(4.99+8.64i)T+(−20.5+35.5i)T2 1 + (4.99 + 8.64i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6.25+10.8i)T+(−21.5−37.2i)T2 1 + (-6.25 + 10.8i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.755+1.30i)T+(−23.5+40.7i)T2 1 + (0.755 + 1.30i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.19−2.07i)T+(−26.5−45.8i)T2 1 + (1.19 - 2.07i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.41+2.44i)T+(−29.5+51.0i)T2 1 + (1.41 + 2.44i)T + (-29.5 + 51.0i)T^{2}
61 1+4.33T+61T2 1 + 4.33T + 61T^{2}
67 1−T+67T2 1 - T + 67T^{2}
71 1+(2−3.46i)T+(−35.5−61.4i)T2 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2}
73 1+(2.16−3.75i)T+(−36.5−63.2i)T2 1 + (2.16 - 3.75i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.30−5.72i)T+(−39.5+68.4i)T2 1 + (-3.30 - 5.72i)T + (-39.5 + 68.4i)T^{2}
83 1+2.82T+83T2 1 + 2.82T + 83T^{2}
89 1+(−3.25+5.63i)T+(−44.5−77.0i)T2 1 + (-3.25 + 5.63i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.83−11.8i)T+(−48.5−84.0i)T2 1 + (6.83 - 11.8i)T + (-48.5 - 84.0i)T^{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

−10.62038186654454957949196322665, −10.30890352033288525190082313754, −9.073285094618360481153253458874, −8.506471401017306110656349133921, −7.06767004553190363969496431811, −6.55923408345641586377799596268, −6.12028226255572705860102289384, −5.21617345683388593996720415946, −3.77814706079153294699345162486, −1.43321711011207616419629326184, 0.59491316310844949994328576582, 1.45489786088699784137537782638, 3.09727931737208258718285322023, 4.39952828842504443741813713425, 5.49249755285042169426601668523, 6.30057653774844438830171170188, 7.71423354379657254041901644199, 8.880872867274206210603272130993, 9.443137761428709148920442236705, 10.15157169401301630181984935412

Graph of the ZZ-function along the critical line