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Properties

Label 2-637-13.3-c1-0-4
Degree 22
Conductor 637637
Sign 0.566−0.824i0.566 - 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: 0.566−0.824i0.566 - 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 0.756708+0.398206i0.756708 + 0.398206i
L(12)L(\frac12) ≈\approx 0.756708+0.398206i0.756708 + 0.398206i
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

−10.82729759026282260547301717660, −9.721405955812685335685363952160, −9.251734122052641190809261776357, −8.324559688547476036254226184074, −7.23831406617797693454834055276, −6.46499156511041531122446872036, −4.74355490065593425920397134226, −3.97218738204525399490237577974, −3.18876440264667660329093342746, −1.45609047296740185031699039074, 0.53232870906208110014845274271, 2.82150506863908742318423780555, 3.57910202067471369815569415453, 4.85820613259197606235553040773, 6.51887812640304525106412077395, 7.04173194717886273713438738542, 8.008515093698000955842782518214, 8.262170893862692574803118450506, 8.981241484820736069845600791414, 10.78179678117143708607293338192

Graph of the ZZ-function along the critical line