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Properties

Label 2-637-91.81-c1-0-21
Degree 22
Conductor 637637
Sign −0.334+0.942i-0.334 + 0.942i
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  + (1.16 − 2.01i)2-s − 2.30·3-s + (−1.71 − 2.97i)4-s + (1.68 + 2.91i)5-s + (−2.69 + 4.66i)6-s − 3.34·8-s + 2.33·9-s + 7.85·10-s + 2.33·11-s + (3.96 + 6.87i)12-s + (−0.408 − 3.58i)13-s + (−3.89 − 6.74i)15-s + (−0.466 + 0.808i)16-s + (−2.72 − 4.72i)17-s + (2.71 − 4.70i)18-s + 7.16·19-s + ⋯
L(s)  = 1  + (0.824 − 1.42i)2-s − 1.33·3-s + (−0.858 − 1.48i)4-s + (0.753 + 1.30i)5-s + (−1.09 + 1.90i)6-s − 1.18·8-s + 0.777·9-s + 2.48·10-s + 0.702·11-s + (1.14 + 1.98i)12-s + (−0.113 − 0.993i)13-s + (−1.00 − 1.74i)15-s + (−0.116 + 0.202i)16-s + (−0.661 − 1.14i)17-s + (0.640 − 1.10i)18-s + 1.64·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.334+0.942i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.334+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.334+0.942i-0.334 + 0.942i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(263,⋅)\chi_{637} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.334+0.942i)(2,\ 637,\ (\ :1/2),\ -0.334 + 0.942i)

Particular Values

L(1)L(1) ≈\approx 0.963578−1.36490i0.963578 - 1.36490i
L(12)L(\frac12) ≈\approx 0.963578−1.36490i0.963578 - 1.36490i
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+(0.408+3.58i)T 1 + (0.408 + 3.58i)T
good2 1+(−1.16+2.01i)T+(−1−1.73i)T2 1 + (-1.16 + 2.01i)T + (-1 - 1.73i)T^{2}
3 1+2.30T+3T2 1 + 2.30T + 3T^{2}
5 1+(−1.68−2.91i)T+(−2.5+4.33i)T2 1 + (-1.68 - 2.91i)T + (-2.5 + 4.33i)T^{2}
11 1−2.33T+11T2 1 - 2.33T + 11T^{2}
17 1+(2.72+4.72i)T+(−8.5+14.7i)T2 1 + (2.72 + 4.72i)T + (-8.5 + 14.7i)T^{2}
19 1−7.16T+19T2 1 - 7.16T + 19T^{2}
23 1+(−3.22+5.58i)T+(−11.5−19.9i)T2 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.22−7.31i)T+(−14.5+25.1i)T2 1 + (-4.22 - 7.31i)T + (-14.5 + 25.1i)T^{2}
31 1+(1.52−2.64i)T+(−15.5−26.8i)T2 1 + (1.52 - 2.64i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.52−2.64i)T+(−18.5−32.0i)T2 1 + (1.52 - 2.64i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.468+0.812i)T+(−20.5+35.5i)T2 1 + (0.468 + 0.812i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.04+3.54i)T+(−21.5−37.2i)T2 1 + (-2.04 + 3.54i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.73+2.99i)T+(−23.5+40.7i)T2 1 + (1.73 + 2.99i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.17+2.02i)T+(−26.5−45.8i)T2 1 + (-1.17 + 2.02i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.62−6.27i)T+(−29.5+51.0i)T2 1 + (-3.62 - 6.27i)T + (-29.5 + 51.0i)T^{2}
61 1+6.39T+61T2 1 + 6.39T + 61T^{2}
67 1−4.61T+67T2 1 - 4.61T + 67T^{2}
71 1+(−3.79+6.57i)T+(−35.5−61.4i)T2 1 + (-3.79 + 6.57i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.03−1.79i)T+(−36.5−63.2i)T2 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.79−6.57i)T+(−39.5+68.4i)T2 1 + (-3.79 - 6.57i)T + (-39.5 + 68.4i)T^{2}
83 1−2.89T+83T2 1 - 2.89T + 83T^{2}
89 1+(−6.57+11.3i)T+(−44.5−77.0i)T2 1 + (-6.57 + 11.3i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.77−3.08i)T+(−48.5−84.0i)T2 1 + (1.77 - 3.08i)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

−10.57751579351862003783821449188, −10.13952808625822453933590814536, −9.130999786823667295281224008166, −7.15276989680948597051378599076, −6.49798157271034221820574154835, −5.39889431925312597316156818598, −4.90903106048617149655649283161, −3.33483290317477363344109664211, −2.61810552408911526246420811017, −0.999022790644618120248558795987, 1.36717629302354125058642966541, 4.02264982061882533249235558985, 4.82333906129287647714677481585, 5.50702901944194747993822828636, 6.13343235579157349847329058936, 6.83588779536854595488423847752, 7.965003042393469732577689487609, 9.037865992694015132218765327021, 9.694373552114434014050155315833, 11.13981378135211865930967893106

Graph of the ZZ-function along the critical line