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Properties

Label 2-637-13.9-c1-0-10
Degree 22
Conductor 637637
Sign −0.999−0.0251i-0.999 - 0.0251i
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  + (−1.21 + 2.10i)2-s + (−0.376 + 0.652i)3-s + (−1.95 − 3.39i)4-s + 0.341·5-s + (−0.916 − 1.58i)6-s + 4.65·8-s + (1.21 + 2.10i)9-s + (−0.415 + 0.719i)10-s + (1.21 − 2.10i)11-s + 2.95·12-s + (2.50 + 2.59i)13-s + (−0.128 + 0.222i)15-s + (−1.74 + 3.02i)16-s + (0.974 + 1.68i)17-s − 5.91·18-s + (3.14 + 5.44i)19-s + ⋯
L(s)  = 1  + (−0.859 + 1.48i)2-s + (−0.217 + 0.376i)3-s + (−0.978 − 1.69i)4-s + 0.152·5-s + (−0.374 − 0.647i)6-s + 1.64·8-s + (0.405 + 0.702i)9-s + (−0.131 + 0.227i)10-s + (0.366 − 0.635i)11-s + 0.851·12-s + (0.693 + 0.720i)13-s + (−0.0332 + 0.0575i)15-s + (−0.437 + 0.757i)16-s + (0.236 + 0.409i)17-s − 1.39·18-s + (0.721 + 1.24i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.999−0.0251i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.999−0.0251i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.999−0.0251i-0.999 - 0.0251i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(295,⋅)\chi_{637} (295, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.999−0.0251i)(2,\ 637,\ (\ :1/2),\ -0.999 - 0.0251i)

Particular Values

L(1)L(1) ≈\approx 0.0102661+0.817542i0.0102661 + 0.817542i
L(12)L(\frac12) ≈\approx 0.0102661+0.817542i0.0102661 + 0.817542i
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+(−2.50−2.59i)T 1 + (-2.50 - 2.59i)T
good2 1+(1.21−2.10i)T+(−1−1.73i)T2 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2}
3 1+(0.376−0.652i)T+(−1.5−2.59i)T2 1 + (0.376 - 0.652i)T + (-1.5 - 2.59i)T^{2}
5 1−0.341T+5T2 1 - 0.341T + 5T^{2}
11 1+(−1.21+2.10i)T+(−5.5−9.52i)T2 1 + (-1.21 + 2.10i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.974−1.68i)T+(−8.5+14.7i)T2 1 + (-0.974 - 1.68i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.14−5.44i)T+(−9.5+16.4i)T2 1 + (-3.14 - 5.44i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.84+3.19i)T+(−11.5−19.9i)T2 1 + (-1.84 + 3.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.22−3.84i)T+(−14.5−25.1i)T2 1 + (2.22 - 3.84i)T + (-14.5 - 25.1i)T^{2}
31 1+1.97T+31T2 1 + 1.97T + 31T^{2}
37 1+(−4.81+8.33i)T+(−18.5−32.0i)T2 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2}
41 1+(6.26−10.8i)T+(−20.5−35.5i)T2 1 + (6.26 - 10.8i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.20−7.28i)T+(−21.5+37.2i)T2 1 + (-4.20 - 7.28i)T + (-21.5 + 37.2i)T^{2}
47 1+9.00T+47T2 1 + 9.00T + 47T^{2}
53 1−1.49T+53T2 1 - 1.49T + 53T^{2}
59 1+(0.313+0.542i)T+(−29.5+51.0i)T2 1 + (0.313 + 0.542i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.571−0.990i)T+(−30.5+52.8i)T2 1 + (-0.571 - 0.990i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.79+4.84i)T+(−33.5−58.0i)T2 1 + (-2.79 + 4.84i)T + (-33.5 - 58.0i)T^{2}
71 1+(4.74+8.22i)T+(−35.5+61.4i)T2 1 + (4.74 + 8.22i)T + (-35.5 + 61.4i)T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−1.41T+83T2 1 - 1.41T + 83T^{2}
89 1+(6.22−10.7i)T+(−44.5−77.0i)T2 1 + (6.22 - 10.7i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.13+8.90i)T+(−48.5+84.0i)T2 1 + (5.13 + 8.90i)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

−10.71340135218825846508186972471, −9.794143783776285042972786072993, −9.223482376379168575480517408679, −8.186273752422438405350178877444, −7.67504114297580053763741108412, −6.50675026437480524934834370217, −5.89582806615182201394609405300, −4.97158100281800306629645489461, −3.74731159138312938036909757254, −1.46984545366465835967030062283, 0.68533658273075388308636848416, 1.80515769542522829470395819796, 3.12516346707865008786577864964, 4.04034477824557599164036466264, 5.50841613685318383193933388313, 6.83605521799790520831705036074, 7.70159368046002348768788674314, 8.774109107701047404091499646453, 9.587215429734493864531662314111, 9.994076583383720241507836044818

Graph of the ZZ-function along the critical line