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Properties

Label 2-637-13.3-c1-0-18
Degree 22
Conductor 637637
Sign 0.973+0.227i0.973 + 0.227i
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  + (0.425 + 0.737i)2-s + (0.330 + 0.572i)3-s + (0.637 − 1.10i)4-s − 3.44·5-s + (−0.281 + 0.487i)6-s + 2.78·8-s + (1.28 − 2.21i)9-s + (−1.46 − 2.53i)10-s + (0.448 + 0.777i)11-s + 0.843·12-s + (3.07 − 1.88i)13-s + (−1.13 − 1.97i)15-s + (−0.0891 − 0.154i)16-s + (0.968 − 1.67i)17-s + 2.18·18-s + (0.519 − 0.898i)19-s + ⋯
L(s)  = 1  + (0.300 + 0.521i)2-s + (0.190 + 0.330i)3-s + (0.318 − 0.552i)4-s − 1.53·5-s + (−0.114 + 0.198i)6-s + 0.985·8-s + (0.427 − 0.739i)9-s + (−0.463 − 0.802i)10-s + (0.135 + 0.234i)11-s + 0.243·12-s + (0.852 − 0.522i)13-s + (−0.293 − 0.508i)15-s + (−0.0222 − 0.0386i)16-s + (0.234 − 0.406i)17-s + 0.514·18-s + (0.119 − 0.206i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.973+0.227i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.973+0.227i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.973+0.227i0.973 + 0.227i
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), 0.973+0.227i)(2,\ 637,\ (\ :1/2),\ 0.973 + 0.227i)

Particular Values

L(1)L(1) ≈\approx 1.69204−0.195193i1.69204 - 0.195193i
L(12)L(\frac12) ≈\approx 1.69204−0.195193i1.69204 - 0.195193i
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+(−3.07+1.88i)T 1 + (-3.07 + 1.88i)T
good2 1+(−0.425−0.737i)T+(−1+1.73i)T2 1 + (-0.425 - 0.737i)T + (-1 + 1.73i)T^{2}
3 1+(−0.330−0.572i)T+(−1.5+2.59i)T2 1 + (-0.330 - 0.572i)T + (-1.5 + 2.59i)T^{2}
5 1+3.44T+5T2 1 + 3.44T + 5T^{2}
11 1+(−0.448−0.777i)T+(−5.5+9.52i)T2 1 + (-0.448 - 0.777i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.968+1.67i)T+(−8.5−14.7i)T2 1 + (-0.968 + 1.67i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.519+0.898i)T+(−9.5−16.4i)T2 1 + (-0.519 + 0.898i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.82+4.89i)T+(−11.5+19.9i)T2 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.917−1.58i)T+(−14.5+25.1i)T2 1 + (-0.917 - 1.58i)T + (-14.5 + 25.1i)T^{2}
31 1−9.13T+31T2 1 - 9.13T + 31T^{2}
37 1+(−5.30−9.17i)T+(−18.5+32.0i)T2 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.66+4.61i)T+(−20.5+35.5i)T2 1 + (2.66 + 4.61i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1.95+3.39i)T+(−21.5−37.2i)T2 1 + (-1.95 + 3.39i)T + (-21.5 - 37.2i)T^{2}
47 1+7.19T+47T2 1 + 7.19T + 47T^{2}
53 1+9.38T+53T2 1 + 9.38T + 53T^{2}
59 1+(0.255−0.442i)T+(−29.5−51.0i)T2 1 + (0.255 - 0.442i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.718+1.24i)T+(−30.5−52.8i)T2 1 + (-0.718 + 1.24i)T + (-30.5 - 52.8i)T^{2}
67 1+(−4.22−7.31i)T+(−33.5+58.0i)T2 1 + (-4.22 - 7.31i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.72+2.98i)T+(−35.5−61.4i)T2 1 + (-1.72 + 2.98i)T + (-35.5 - 61.4i)T^{2}
73 1+10.9T+73T2 1 + 10.9T + 73T^{2}
79 1+12.0T+79T2 1 + 12.0T + 79T^{2}
83 1+1.51T+83T2 1 + 1.51T + 83T^{2}
89 1+(−6.80−11.7i)T+(−44.5+77.0i)T2 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.253+0.438i)T+(−48.5−84.0i)T2 1 + (-0.253 + 0.438i)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

−10.50802584959899506452063781085, −9.826133593007085327221591238358, −8.554094582872295498404525104265, −7.88147879205476349757075308948, −6.89060962199816916086229275611, −6.23824050031603354714177891347, −4.82273011956770474829396825167, −4.15836346654335059506933594316, −3.07634512198213374880369624647, −0.949522021807354355401760335051, 1.54195222339506745114845247166, 3.01331195750254048415041859487, 3.92912413072989850208013847787, 4.58204965723302082594607634166, 6.28363043479190534648720001033, 7.38445748707813653936363781759, 7.932873295874432540146230573045, 8.477775860076572967741699863455, 9.957284493029775115233486073852, 11.07094532380806159396062954222

Graph of the ZZ-function along the critical line