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Properties

Label 2-637-7.4-c1-0-37
Degree 22
Conductor 637637
Sign −0.605−0.795i-0.605 - 0.795i
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  + (−0.707 − 1.22i)2-s + (0.707 − 1.22i)3-s + (−2.20 − 3.82i)5-s − 2·6-s − 2.82·8-s + (0.500 + 0.866i)9-s + (−3.12 + 5.40i)10-s + (2.12 − 3.67i)11-s − 13-s − 6.24·15-s + (2.00 + 3.46i)16-s + (0.707 − 1.22i)17-s + (0.707 − 1.22i)18-s + (−0.621 − 1.07i)19-s − 6·22-s + (0.0857 + 0.148i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (0.408 − 0.707i)3-s + (−0.987 − 1.70i)5-s − 0.816·6-s − 0.999·8-s + (0.166 + 0.288i)9-s + (−0.987 + 1.70i)10-s + (0.639 − 1.10i)11-s − 0.277·13-s − 1.61·15-s + (0.500 + 0.866i)16-s + (0.171 − 0.297i)17-s + (0.166 − 0.288i)18-s + (−0.142 − 0.246i)19-s − 1.27·22-s + (0.0178 + 0.0309i)23-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.605−0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.605−0.795i-0.605 - 0.795i
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), −0.605−0.795i)(2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)

Particular Values

L(1)L(1) ≈\approx 0.435477+0.878461i0.435477 + 0.878461i
L(12)L(\frac12) ≈\approx 0.435477+0.878461i0.435477 + 0.878461i
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+(0.707+1.22i)T+(−1+1.73i)T2 1 + (0.707 + 1.22i)T + (-1 + 1.73i)T^{2}
3 1+(−0.707+1.22i)T+(−1.5−2.59i)T2 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2}
5 1+(2.20+3.82i)T+(−2.5+4.33i)T2 1 + (2.20 + 3.82i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.12+3.67i)T+(−5.5−9.52i)T2 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.707+1.22i)T+(−8.5−14.7i)T2 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.621+1.07i)T+(−9.5+16.4i)T2 1 + (0.621 + 1.07i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.0857−0.148i)T+(−11.5+19.9i)T2 1 + (-0.0857 - 0.148i)T + (-11.5 + 19.9i)T^{2}
29 1−5.82T+29T2 1 - 5.82T + 29T^{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+(−3.12−5.40i)T+(−18.5+32.0i)T2 1 + (-3.12 - 5.40i)T + (-18.5 + 32.0i)T^{2}
41 1−3.17T+41T2 1 - 3.17T + 41T^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1+(2.20+3.82i)T+(−23.5+40.7i)T2 1 + (2.20 + 3.82i)T + (-23.5 + 40.7i)T^{2}
53 1+(−2.91+5.04i)T+(−26.5−45.8i)T2 1 + (-2.91 + 5.04i)T + (-26.5 - 45.8i)T^{2}
59 1+(5.82−10.0i)T+(−29.5−51.0i)T2 1 + (5.82 - 10.0i)T + (-29.5 - 51.0i)T^{2}
61 1+(3+5.19i)T+(−30.5+52.8i)T2 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.24−2.15i)T+(−33.5−58.0i)T2 1 + (1.24 - 2.15i)T + (-33.5 - 58.0i)T^{2}
71 1−1.07T+71T2 1 - 1.07T + 71T^{2}
73 1+(−0.378+0.655i)T+(−36.5−63.2i)T2 1 + (-0.378 + 0.655i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.742−1.28i)T+(−39.5+68.4i)T2 1 + (-0.742 - 1.28i)T + (-39.5 + 68.4i)T^{2}
83 1−4.75T+83T2 1 - 4.75T + 83T^{2}
89 1+(2.20+3.82i)T+(−44.5+77.0i)T2 1 + (2.20 + 3.82i)T + (-44.5 + 77.0i)T^{2}
97 1+13.7T+97T2 1 + 13.7T + 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

−9.911611071121596212563920079693, −9.013031209766180282432741745332, −8.430769288923763980473233015533, −7.82033781167227163561025773279, −6.52703594860114808257873580176, −5.28555950376612400791514332397, −4.21938404747790318117721286375, −2.94196093513247449332513168048, −1.48219492623847272883945898622, −0.63924813676598929352401460753, 2.70969450347938496056195857568, 3.57050807769704823976783886776, 4.43785922933592142099961430160, 6.31221564393132257067580687216, 6.83678929496537229541908280635, 7.55185644320047912807710415014, 8.330664170087626497799734612701, 9.405894285816465110330029308166, 10.10564897225060019790625778978, 10.90560348233766062819855495140

Graph of the ZZ-function along the critical line