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Properties

Label 2-637-91.74-c1-0-12
Degree 22
Conductor 637637
Sign 0.894+0.446i0.894 + 0.446i
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  − 1.90·2-s + (0.214 + 0.371i)3-s + 1.63·4-s + (−0.736 − 1.27i)5-s + (−0.408 − 0.707i)6-s + 0.702·8-s + (1.40 − 2.43i)9-s + (1.40 + 2.43i)10-s + (2.19 + 3.80i)11-s + (0.349 + 0.605i)12-s + (−2.69 + 2.39i)13-s + (0.315 − 0.546i)15-s − 4.60·16-s + 1.20·17-s + (−2.68 + 4.64i)18-s + (1.62 − 2.80i)19-s + ⋯
L(s)  = 1  − 1.34·2-s + (0.123 + 0.214i)3-s + 0.815·4-s + (−0.329 − 0.570i)5-s + (−0.166 − 0.288i)6-s + 0.248·8-s + (0.469 − 0.813i)9-s + (0.443 + 0.768i)10-s + (0.662 + 1.14i)11-s + (0.100 + 0.174i)12-s + (−0.748 + 0.663i)13-s + (0.0814 − 0.141i)15-s − 1.15·16-s + 0.291·17-s + (−0.632 + 1.09i)18-s + (0.371 − 0.644i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.894+0.446i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.894+0.446i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.894+0.446i0.894 + 0.446i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(165,⋅)\chi_{637} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.894+0.446i)(2,\ 637,\ (\ :1/2),\ 0.894 + 0.446i)

Particular Values

L(1)L(1) ≈\approx 0.705747−0.166416i0.705747 - 0.166416i
L(12)L(\frac12) ≈\approx 0.705747−0.166416i0.705747 - 0.166416i
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+(2.69−2.39i)T 1 + (2.69 - 2.39i)T
good2 1+1.90T+2T2 1 + 1.90T + 2T^{2}
3 1+(−0.214−0.371i)T+(−1.5+2.59i)T2 1 + (-0.214 - 0.371i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.736+1.27i)T+(−2.5+4.33i)T2 1 + (0.736 + 1.27i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.19−3.80i)T+(−5.5+9.52i)T2 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2}
17 1−1.20T+17T2 1 - 1.20T + 17T^{2}
19 1+(−1.62+2.80i)T+(−9.5−16.4i)T2 1 + (-1.62 + 2.80i)T + (-9.5 - 16.4i)T^{2}
23 1+4.43T+23T2 1 + 4.43T + 23T^{2}
29 1+(0.0837−0.145i)T+(−14.5−25.1i)T2 1 + (0.0837 - 0.145i)T + (-14.5 - 25.1i)T^{2}
31 1+(−2.62+4.54i)T+(−15.5−26.8i)T2 1 + (-2.62 + 4.54i)T + (-15.5 - 26.8i)T^{2}
37 1−7.05T+37T2 1 - 7.05T + 37T^{2}
41 1+(−2.58+4.47i)T+(−20.5−35.5i)T2 1 + (-2.58 + 4.47i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.0113+0.0197i)T+(−21.5+37.2i)T2 1 + (0.0113 + 0.0197i)T + (-21.5 + 37.2i)T^{2}
47 1+(−5.84−10.1i)T+(−23.5+40.7i)T2 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2}
53 1+(−0.0708+0.122i)T+(−26.5−45.8i)T2 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2}
59 1−5.34T+59T2 1 - 5.34T + 59T^{2}
61 1+(−5.77+9.99i)T+(−30.5−52.8i)T2 1 + (-5.77 + 9.99i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.06+3.58i)T+(−33.5+58.0i)T2 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2}
71 1+(−4.98−8.63i)T+(−35.5+61.4i)T2 1 + (-4.98 - 8.63i)T + (-35.5 + 61.4i)T^{2}
73 1+(−7.62+13.1i)T+(−36.5−63.2i)T2 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.387+0.670i)T+(−39.5+68.4i)T2 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2}
83 1−16.0T+83T2 1 - 16.0T + 83T^{2}
89 1+6.55T+89T2 1 + 6.55T + 89T^{2}
97 1+(−1.74−3.02i)T+(−48.5+84.0i)T2 1 + (-1.74 - 3.02i)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

−10.01825089649615216348260010088, −9.543687523807550204171028763281, −9.074059497492519170512900797581, −7.995822395142424096115357289495, −7.26454040672391785057778040010, −6.43297120665741401742128476786, −4.68533012903465507393063992099, −4.11072420633955706857468804404, −2.18762863127060956220944978898, −0.811997872974530704723468565031, 1.06037455747608530093793971085, 2.52479101997313739219425835019, 3.86071348566735309624975574246, 5.28438768790229430177068411622, 6.55954873074569235649931196735, 7.51138206892934270889990625400, 7.993593341207264705586625776028, 8.807797872724497893207643247131, 9.890260707294501151215251783050, 10.39224013085585195437956401213

Graph of the ZZ-function along the critical line