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Properties

Label 2-637-13.3-c1-0-27
Degree 22
Conductor 637637
Sign −0.566+0.824i-0.566 + 0.824i
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  + (−0.5 − 0.866i)2-s + (−0.707 − 1.22i)3-s + (0.500 − 0.866i)4-s + 4.09·5-s + (−0.707 + 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−2.04 − 3.54i)10-s + (1.89 + 3.28i)11-s − 1.41·12-s + (−0.634 − 3.54i)13-s + (−2.89 − 5.01i)15-s + (0.500 + 0.866i)16-s + (0.634 − 1.09i)17-s − 1.00·18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.408 − 0.707i)3-s + (0.250 − 0.433i)4-s + 1.83·5-s + (−0.288 + 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.647 − 1.12i)10-s + (0.572 + 0.991i)11-s − 0.408·12-s + (−0.176 − 0.984i)13-s + (−0.748 − 1.29i)15-s + (0.125 + 0.216i)16-s + (0.153 − 0.266i)17-s − 0.235·18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.566+0.824i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.566+0.824i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.566+0.824i-0.566 + 0.824i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(393,⋅)\chi_{637} (393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.566+0.824i)(2,\ 637,\ (\ :1/2),\ -0.566 + 0.824i)

Particular Values

L(1)L(1) ≈\approx 0.746473−1.41851i0.746473 - 1.41851i
L(12)L(\frac12) ≈\approx 0.746473−1.41851i0.746473 - 1.41851i
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+(0.634+3.54i)T 1 + (0.634 + 3.54i)T
good2 1+(0.5+0.866i)T+(−1+1.73i)T2 1 + (0.5 + 0.866i)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 1−4.09T+5T2 1 - 4.09T + 5T^{2}
11 1+(−1.89−3.28i)T+(−5.5+9.52i)T2 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.634+1.09i)T+(−8.5−14.7i)T2 1 + (-0.634 + 1.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.41+2.44i)T+(−9.5−16.4i)T2 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.89−6.75i)T+(−11.5+19.9i)T2 1 + (-3.89 - 6.75i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.397+0.689i)T+(−14.5+25.1i)T2 1 + (0.397 + 0.689i)T + (-14.5 + 25.1i)T^{2}
31 1+1.41T+31T2 1 + 1.41T + 31T^{2}
37 1+(1.39+2.42i)T+(−18.5+32.0i)T2 1 + (1.39 + 2.42i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.48+2.57i)T+(−20.5+35.5i)T2 1 + (1.48 + 2.57i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.89−6.75i)T+(−21.5−37.2i)T2 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+12.5T+53T2 1 + 12.5T + 53T^{2}
59 1+(−6.21+10.7i)T+(−29.5−51.0i)T2 1 + (-6.21 + 10.7i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.17−7.22i)T+(−30.5−52.8i)T2 1 + (4.17 - 7.22i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.89−3.28i)T+(−33.5+58.0i)T2 1 + (-1.89 - 3.28i)T + (-33.5 + 58.0i)T^{2}
71 1+(3−5.19i)T+(−35.5−61.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1+12.5T+73T2 1 + 12.5T + 73T^{2}
79 1+2.20T+79T2 1 + 2.20T + 79T^{2}
83 1+9.89T+83T2 1 + 9.89T + 83T^{2}
89 1+(−7.48−12.9i)T+(−44.5+77.0i)T2 1 + (-7.48 - 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1+(−2.12+3.67i)T+(−48.5−84.0i)T2 1 + (-2.12 + 3.67i)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

−10.08662103297143632092617027769, −9.620886269399728999300259177659, −9.073796135688037548811002529183, −7.33264529646235685025673298359, −6.62565652719221107625113687229, −5.80302163376652866964315317000, −5.13515405868487536186666554242, −3.03905272948381252521085574597, −1.88022673012806132182412890956, −1.14423458857019977662047049032, 1.83045509365056935015039331927, 3.17073437718250360058143175614, 4.63446522518492835427187290604, 5.72289784736615186666225095090, 6.29832997321078312210044553898, 7.12671152473831952617863435905, 8.540418093346059376237589206280, 9.106600271988748835685429285810, 9.919787923764635484117936017566, 10.67071817006721559873442472223

Graph of the ZZ-function along the critical line