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Properties

Label 2-650-65.37-c1-0-17
Degree 22
Conductor 650650
Sign −0.939+0.342i-0.939 + 0.342i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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  + (−0.866 + 0.5i)2-s + (−1.09 − 0.294i)3-s + (0.499 − 0.866i)4-s + (1.09 − 0.294i)6-s + (2.54 − 4.40i)7-s + 0.999i·8-s + (−1.47 − 0.854i)9-s + (−2.27 − 0.609i)11-s + (−0.803 + 0.803i)12-s + (−3.54 − 0.653i)13-s + 5.08i·14-s + (−0.5 − 0.866i)16-s + (0.0601 + 0.224i)17-s + 1.70·18-s + (1.53 + 5.71i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.633 − 0.169i)3-s + (0.249 − 0.433i)4-s + (0.448 − 0.120i)6-s + (0.961 − 1.66i)7-s + 0.353i·8-s + (−0.493 − 0.284i)9-s + (−0.685 − 0.183i)11-s + (−0.231 + 0.231i)12-s + (−0.983 − 0.181i)13-s + 1.35i·14-s + (−0.125 − 0.216i)16-s + (0.0145 + 0.0544i)17-s + 0.402·18-s + (0.351 + 1.31i)19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.939+0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.939+0.342i-0.939 + 0.342i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(557,⋅)\chi_{650} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.939+0.342i)(2,\ 650,\ (\ :1/2),\ -0.939 + 0.342i)

Particular Values

L(1)L(1) ≈\approx 0.0542787−0.306960i0.0542787 - 0.306960i
L(12)L(\frac12) ≈\approx 0.0542787−0.306960i0.0542787 - 0.306960i
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)
bad2 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
5 1 1
13 1+(3.54+0.653i)T 1 + (3.54 + 0.653i)T
good3 1+(1.09+0.294i)T+(2.59+1.5i)T2 1 + (1.09 + 0.294i)T + (2.59 + 1.5i)T^{2}
7 1+(−2.54+4.40i)T+(−3.5−6.06i)T2 1 + (-2.54 + 4.40i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.27+0.609i)T+(9.52+5.5i)T2 1 + (2.27 + 0.609i)T + (9.52 + 5.5i)T^{2}
17 1+(−0.0601−0.224i)T+(−14.7+8.5i)T2 1 + (-0.0601 - 0.224i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.53−5.71i)T+(−16.4+9.5i)T2 1 + (-1.53 - 5.71i)T + (-16.4 + 9.5i)T^{2}
23 1+(0.674−2.51i)T+(−19.9−11.5i)T2 1 + (0.674 - 2.51i)T + (-19.9 - 11.5i)T^{2}
29 1+(2.64−1.52i)T+(14.5−25.1i)T2 1 + (2.64 - 1.52i)T + (14.5 - 25.1i)T^{2}
31 1+(4.45+4.45i)T+31iT2 1 + (4.45 + 4.45i)T + 31iT^{2}
37 1+(2.49+4.31i)T+(−18.5+32.0i)T2 1 + (2.49 + 4.31i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.412−1.53i)T+(−35.5−20.5i)T2 1 + (0.412 - 1.53i)T + (-35.5 - 20.5i)T^{2}
43 1+(1.66−0.447i)T+(37.2−21.5i)T2 1 + (1.66 - 0.447i)T + (37.2 - 21.5i)T^{2}
47 1+12.2T+47T2 1 + 12.2T + 47T^{2}
53 1+(5.79−5.79i)T−53iT2 1 + (5.79 - 5.79i)T - 53iT^{2}
59 1+(2.68−0.720i)T+(51.0−29.5i)T2 1 + (2.68 - 0.720i)T + (51.0 - 29.5i)T^{2}
61 1+(−4.23+7.33i)T+(−30.5−52.8i)T2 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.150−0.0866i)T+(33.5−58.0i)T2 1 + (0.150 - 0.0866i)T + (33.5 - 58.0i)T^{2}
71 1+(−4.17+1.11i)T+(61.4−35.5i)T2 1 + (-4.17 + 1.11i)T + (61.4 - 35.5i)T^{2}
73 1+7.51iT−73T2 1 + 7.51iT - 73T^{2}
79 1−5.17iT−79T2 1 - 5.17iT - 79T^{2}
83 1−10.4T+83T2 1 - 10.4T + 83T^{2}
89 1+(−1.49+5.59i)T+(−77.0−44.5i)T2 1 + (-1.49 + 5.59i)T + (-77.0 - 44.5i)T^{2}
97 1+(−7.02−4.05i)T+(48.5+84.0i)T2 1 + (-7.02 - 4.05i)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

−10.27058273699329392257004160943, −9.430615976513263840028970122677, −7.995008060268446715229801747507, −7.71808010727264829210805679104, −6.78104767369666507467219436821, −5.64239051375283223029398371040, −4.88363399651925718654090291445, −3.53586096449382512429720887694, −1.63945699087803674788010051372, −0.21382011072114182108978444474, 2.06000749279195726190727307282, 2.84092410066524580278693924012, 5.00209795849935858880149515756, 5.14196868209246119830969772146, 6.46605955890926345867651637610, 7.67305739000469605666099103095, 8.475884353214773168234702113780, 9.160257466920681167882654671115, 10.12043223724868710451209618882, 11.11287477232795605796382691363

Graph of the ZZ-function along the critical line