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Properties

Label 2-637-13.3-c1-0-41
Degree 22
Conductor 637637
Sign 0.872−0.488i0.872 - 0.488i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.5 − 2.59i)3-s + (0.500 − 0.866i)4-s − 3·5-s + (−1.5 + 2.59i)6-s − 3·8-s + (−3 + 5.19i)9-s + (1.5 + 2.59i)10-s + (1.5 + 2.59i)11-s − 3·12-s + (1 − 3.46i)13-s + (4.5 + 7.79i)15-s + (0.500 + 0.866i)16-s + (−1 + 1.73i)17-s + 6·18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 − 1.49i)3-s + (0.250 − 0.433i)4-s − 1.34·5-s + (−0.612 + 1.06i)6-s − 1.06·8-s + (−1 + 1.73i)9-s + (0.474 + 0.821i)10-s + (0.452 + 0.783i)11-s − 0.866·12-s + (0.277 − 0.960i)13-s + (1.16 + 2.01i)15-s + (0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + 1.41·18-s + (−0.114 + 0.198i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.872−0.488i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.872−0.488i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.872−0.488i0.872 - 0.488i
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: 11
Selberg data: (2, 637, ( :1/2), 0.872−0.488i)(2,\ 637,\ (\ :1/2),\ 0.872 - 0.488i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(−1+3.46i)T 1 + (-1 + 3.46i)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+(1.5+2.59i)T+(−1.5+2.59i)T2 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
11 1+(−1.5−2.59i)T+(−5.5+9.52i)T2 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2}
17 1+(1−1.73i)T+(−8.5−14.7i)T2 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.5−0.866i)T+(−9.5−16.4i)T2 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(3.5+6.06i)T+(−14.5+25.1i)T2 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1+(1+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.5−2.59i)T+(−20.5+35.5i)T2 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.5+6.06i)T+(−21.5−37.2i)T2 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}
47 1+T+47T2 1 + T + 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1+(2−3.46i)T+(−29.5−51.0i)T2 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.5−11.2i)T+(−30.5−52.8i)T2 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.5−2.59i)T+(−33.5+58.0i)T2 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2}
71 1+(6.5−11.2i)T+(−35.5−61.4i)T2 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2}
73 1−13T+73T2 1 - 13T + 73T^{2}
79 1+3T+79T2 1 + 3T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−3−5.19i)T+(−44.5+77.0i)T2 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.5−4.33i)T+(−48.5−84.0i)T2 1 + (2.5 - 4.33i)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

−10.11119986112905739488476531203, −8.782157788270012433070236009096, −7.79567633373124016427401213919, −7.20892396848169732998825106118, −6.26385727754423665528886900574, −5.46005584618270116139243791811, −3.94114065870228253874817188542, −2.39667627224644028421078766268, −1.18585911043870389244667343960, 0, 3.33453984966480936188307050410, 3.91578565359133822835240984392, 4.89627513052537269191172363045, 6.08453405277062383756315652768, 6.89395633091701794443273327264, 7.926414403291826510603158269533, 8.953065874571569495482158931145, 9.319309177067364670683997106943, 10.80761314769465912166222247823

Graph of the ZZ-function along the critical line