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Properties

Label 2-637-91.74-c1-0-5
Degree 22
Conductor 637637
Sign −0.835−0.548i-0.835 - 0.548i
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  − 2.33·2-s + (1.15 + 1.99i)3-s + 3.43·4-s + (1.68 + 2.91i)5-s + (−2.69 − 4.66i)6-s − 3.34·8-s + (−1.16 + 2.01i)9-s + (−3.92 − 6.80i)10-s + (−1.16 − 2.01i)11-s + (3.96 + 6.87i)12-s + (−0.408 + 3.58i)13-s + (−3.89 + 6.74i)15-s + 0.933·16-s + 5.45·17-s + (2.71 − 4.70i)18-s + (−3.58 + 6.20i)19-s + ⋯
L(s)  = 1  − 1.64·2-s + (0.666 + 1.15i)3-s + 1.71·4-s + (0.753 + 1.30i)5-s + (−1.09 − 1.90i)6-s − 1.18·8-s + (−0.388 + 0.673i)9-s + (−1.24 − 2.15i)10-s + (−0.351 − 0.608i)11-s + (1.14 + 1.98i)12-s + (−0.113 + 0.993i)13-s + (−1.00 + 1.74i)15-s + 0.233·16-s + 1.32·17-s + (0.640 − 1.10i)18-s + (−0.822 + 1.42i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.835−0.548i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.835−0.548i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.835−0.548i-0.835 - 0.548i
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), −0.835−0.548i)(2,\ 637,\ (\ :1/2),\ -0.835 - 0.548i)

Particular Values

L(1)L(1) ≈\approx 0.252451+0.844441i0.252451 + 0.844441i
L(12)L(\frac12) ≈\approx 0.252451+0.844441i0.252451 + 0.844441i
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+(0.408−3.58i)T 1 + (0.408 - 3.58i)T
good2 1+2.33T+2T2 1 + 2.33T + 2T^{2}
3 1+(−1.15−1.99i)T+(−1.5+2.59i)T2 1 + (-1.15 - 1.99i)T + (-1.5 + 2.59i)T^{2}
5 1+(−1.68−2.91i)T+(−2.5+4.33i)T2 1 + (-1.68 - 2.91i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.16+2.01i)T+(−5.5+9.52i)T2 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2}
17 1−5.45T+17T2 1 - 5.45T + 17T^{2}
19 1+(3.58−6.20i)T+(−9.5−16.4i)T2 1 + (3.58 - 6.20i)T + (-9.5 - 16.4i)T^{2}
23 1+6.45T+23T2 1 + 6.45T + 23T^{2}
29 1+(−4.22+7.31i)T+(−14.5−25.1i)T2 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.52−2.64i)T+(−15.5−26.8i)T2 1 + (1.52 - 2.64i)T + (-15.5 - 26.8i)T^{2}
37 1−3.05T+37T2 1 - 3.05T + 37T^{2}
41 1+(0.468−0.812i)T+(−20.5−35.5i)T2 1 + (0.468 - 0.812i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.04−3.54i)T+(−21.5+37.2i)T2 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.73+2.99i)T+(−23.5+40.7i)T2 1 + (1.73 + 2.99i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.17+2.02i)T+(−26.5−45.8i)T2 1 + (-1.17 + 2.02i)T + (-26.5 - 45.8i)T^{2}
59 1+7.24T+59T2 1 + 7.24T + 59T^{2}
61 1+(−3.19+5.53i)T+(−30.5−52.8i)T2 1 + (-3.19 + 5.53i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.30+3.99i)T+(−33.5+58.0i)T2 1 + (2.30 + 3.99i)T + (-33.5 + 58.0i)T^{2}
71 1+(−3.79−6.57i)T+(−35.5+61.4i)T2 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2}
73 1+(1.03−1.79i)T+(−36.5−63.2i)T2 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.79−6.57i)T+(−39.5+68.4i)T2 1 + (-3.79 - 6.57i)T + (-39.5 + 68.4i)T^{2}
83 1−2.89T+83T2 1 - 2.89T + 83T^{2}
89 1+13.1T+89T2 1 + 13.1T + 89T^{2}
97 1+(1.77+3.08i)T+(−48.5+84.0i)T2 1 + (1.77 + 3.08i)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

−10.40674757555282032997452833836, −9.903321990126708379427809311791, −9.588823870713672217835834768173, −8.397817216332858537120344564562, −7.88309219720193776464377792566, −6.64531929480115688973155720272, −5.89271212821711254313855585216, −4.07960432695075579383628748896, −2.96319370209523764930603837497, −1.93736358616088475201393040636, 0.76423655163299994550154811930, 1.72888373508026676429052858210, 2.65695754868378917317304619241, 4.86142367958467943862486740856, 6.04599615811058381188619099883, 7.18909228159948037538798500749, 7.85889507484750269714745716379, 8.481226904589051868692541887940, 9.141913273120150323645932037999, 9.959248288946991702269254996789

Graph of the ZZ-function along the critical line