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Properties

Label 2-637-13.9-c1-0-31
Degree 22
Conductor 637637
Sign −0.0662+0.997i-0.0662 + 0.997i
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  + (0.952 − 1.65i)2-s + (0.214 − 0.371i)3-s + (−0.815 − 1.41i)4-s + 1.47·5-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 − 0.698·12-s + (−2.69 + 2.39i)13-s + (0.315 − 0.546i)15-s + (2.30 − 3.98i)16-s + (−0.601 − 1.04i)17-s + 5.36·18-s + (1.62 + 2.80i)19-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)2-s + (0.123 − 0.214i)3-s + (−0.407 − 0.706i)4-s + 0.658·5-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 − 0.201·12-s + (−0.748 + 0.663i)13-s + (0.0814 − 0.141i)15-s + (0.575 − 0.996i)16-s + (−0.145 − 0.252i)17-s + 1.26·18-s + (0.371 + 0.644i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.0662+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.0662+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.0662+0.997i-0.0662 + 0.997i
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), −0.0662+0.997i)(2,\ 637,\ (\ :1/2),\ -0.0662 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 1.79863−1.92200i1.79863 - 1.92200i
L(12)L(\frac12) ≈\approx 1.79863−1.92200i1.79863 - 1.92200i
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+(−0.952+1.65i)T+(−1−1.73i)T2 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{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 1−1.47T+5T2 1 - 1.47T + 5T^{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+(0.601+1.04i)T+(−8.5+14.7i)T2 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{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+(−2.21+3.84i)T+(−11.5−19.9i)T2 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{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+5.24T+31T2 1 + 5.24T + 31T^{2}
37 1+(3.52−6.10i)T+(−18.5−32.0i)T2 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{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+11.6T+47T2 1 + 11.6T + 47T^{2}
53 1+0.141T+53T2 1 + 0.141T + 53T^{2}
59 1+(2.67+4.62i)T+(−29.5+51.0i)T2 1 + (2.67 + 4.62i)T + (-29.5 + 51.0i)T^{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+15.2T+73T2 1 + 15.2T + 73T^{2}
79 1−0.774T+79T2 1 - 0.774T + 79T^{2}
83 1−16.0T+83T2 1 - 16.0T + 83T^{2}
89 1+(−3.27+5.67i)T+(−44.5−77.0i)T2 1 + (-3.27 + 5.67i)T + (-44.5 - 77.0i)T^{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

−10.47243284934373811217494805799, −9.822541607415890265045814125430, −8.831231251741952447931153830158, −7.70008068916775842636549925169, −6.70284285982844094072486676390, −5.50348382428050391925076611332, −4.60471264221294279352924935556, −3.52613901911939569504501647789, −2.39125458713730210177750496910, −1.46713675594188325938413675365, 1.76103362439323387243986848632, 3.52435549322508809895105736344, 4.59413447888692580824827373955, 5.36913233024785055545320841917, 6.36669834854645623675837926193, 7.08187092447380363169152024318, 7.77785038667649480175568226558, 9.233306508850426564696787788351, 9.668520040290791458923444913427, 10.62497398648654110588637193324

Graph of the ZZ-function along the critical line