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Properties

Label 2-637-91.16-c1-0-28
Degree 22
Conductor 637637
Sign 0.788+0.615i0.788 + 0.615i
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  + 2-s + (−1.5 + 2.59i)3-s − 4-s + (1.5 − 2.59i)5-s + (−1.5 + 2.59i)6-s − 3·8-s + (−3 − 5.19i)9-s + (1.5 − 2.59i)10-s + (1.5 − 2.59i)11-s + (1.5 − 2.59i)12-s + (1 − 3.46i)13-s + (4.5 + 7.79i)15-s − 16-s + 2·17-s + (−3 − 5.19i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 1.49i)3-s − 0.5·4-s + (0.670 − 1.16i)5-s + (−0.612 + 1.06i)6-s − 1.06·8-s + (−1 − 1.73i)9-s + (0.474 − 0.821i)10-s + (0.452 − 0.783i)11-s + (0.433 − 0.749i)12-s + (0.277 − 0.960i)13-s + (1.16 + 2.01i)15-s − 0.250·16-s + 0.485·17-s + (−0.707 − 1.22i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.788+0.615i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.788+0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.788+0.615i0.788 + 0.615i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(471,⋅)\chi_{637} (471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.788+0.615i)(2,\ 637,\ (\ :1/2),\ 0.788 + 0.615i)

Particular Values

L(1)L(1) ≈\approx 1.16211−0.400115i1.16211 - 0.400115i
L(12)L(\frac12) ≈\approx 1.16211−0.400115i1.16211 - 0.400115i
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+3.46i)T 1 + (-1 + 3.46i)T
good2 1−T+2T2 1 - T + 2T^{2}
3 1+(1.5−2.59i)T+(−1.5−2.59i)T2 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.5+2.59i)T+(−2.5−4.33i)T2 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.5+2.59i)T+(−5.5−9.52i)T2 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+(0.5+0.866i)T+(−9.5+16.4i)T2 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2}
23 1+23T2 1 + 23T^{2}
29 1+(3.5+6.06i)T+(−14.5+25.1i)T2 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.5−2.59i)T+(−15.5+26.8i)T2 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(−1.5−2.59i)T+(−20.5+35.5i)T2 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.5+6.06i)T+(−21.5−37.2i)T2 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.5+0.866i)T+(−23.5−40.7i)T2 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.5+2.59i)T+(−26.5+45.8i)T2 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+(6.5+11.2i)T+(−30.5+52.8i)T2 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.5+2.59i)T+(−33.5−58.0i)T2 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2}
71 1+(6.5−11.2i)T+(−35.5−61.4i)T2 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2}
73 1+(6.5+11.2i)T+(−36.5+63.2i)T2 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2}
79 1+(−1.5+2.59i)T+(−39.5−68.4i)T2 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(2.5−4.33i)T+(−48.5−84.0i)T2 1 + (2.5 - 4.33i)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

−10.40862877019752126746322511114, −9.600246175692059040102002772527, −9.069295824135164454064268136349, −8.253635428651120014387623823920, −6.11055666883225270480791372074, −5.64632314701751292668146259107, −4.99440595445804030262982022108, −4.17980600137660020231207138725, −3.28969065644276697809865200440, −0.63879991993342906314309035535, 1.55535963690720593600409619946, 2.75810618453237808642489293905, 4.24172625753363698227480771009, 5.52199942051984384581974102511, 6.19608207106081283978248930988, 6.83558622988435071504855307682, 7.59423588506133414293067071337, 8.949953815024556367006728802541, 9.922556701233750088242751248758, 10.97362943160390685804829381039

Graph of the ZZ-function along the critical line