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Properties

Label 2-637-7.4-c1-0-21
Degree 22
Conductor 637637
Sign 0.947−0.318i0.947 - 0.318i
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  + (0.588 + 1.01i)2-s + (−1.67 + 2.90i)3-s + (0.308 − 0.533i)4-s + (−1.57 − 2.72i)5-s − 3.94·6-s + 3.07·8-s + (−4.11 − 7.12i)9-s + (1.85 − 3.20i)10-s + (0.386 − 0.669i)11-s + (1.03 + 1.78i)12-s + 13-s + 10.5·15-s + (1.19 + 2.06i)16-s + (2.87 − 4.98i)17-s + (4.83 − 8.38i)18-s + (0.611 + 1.05i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.720i)2-s + (−0.967 + 1.67i)3-s + (0.154 − 0.266i)4-s + (−0.704 − 1.21i)5-s − 1.60·6-s + 1.08·8-s + (−1.37 − 2.37i)9-s + (0.585 − 1.01i)10-s + (0.116 − 0.201i)11-s + (0.298 + 0.516i)12-s + 0.277·13-s + 2.72·15-s + (0.298 + 0.516i)16-s + (0.697 − 1.20i)17-s + (1.14 − 1.97i)18-s + (0.140 + 0.242i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.947−0.318i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.947−0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.947−0.318i0.947 - 0.318i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(508,⋅)\chi_{637} (508, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.947−0.318i)(2,\ 637,\ (\ :1/2),\ 0.947 - 0.318i)

Particular Values

L(1)L(1) ≈\approx 1.18869+0.194406i1.18869 + 0.194406i
L(12)L(\frac12) ≈\approx 1.18869+0.194406i1.18869 + 0.194406i
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−T 1 - T
good2 1+(−0.588−1.01i)T+(−1+1.73i)T2 1 + (-0.588 - 1.01i)T + (-1 + 1.73i)T^{2}
3 1+(1.67−2.90i)T+(−1.5−2.59i)T2 1 + (1.67 - 2.90i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.57+2.72i)T+(−2.5+4.33i)T2 1 + (1.57 + 2.72i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.386+0.669i)T+(−5.5−9.52i)T2 1 + (-0.386 + 0.669i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.87+4.98i)T+(−8.5−14.7i)T2 1 + (-2.87 + 4.98i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.611−1.05i)T+(−9.5+16.4i)T2 1 + (-0.611 - 1.05i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.49−2.59i)T+(−11.5+19.9i)T2 1 + (-1.49 - 2.59i)T + (-11.5 + 19.9i)T^{2}
29 1+2.46T+29T2 1 + 2.46T + 29T^{2}
31 1+(−3.06+5.31i)T+(−15.5−26.8i)T2 1 + (-3.06 + 5.31i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.49+4.32i)T+(−18.5+32.0i)T2 1 + (2.49 + 4.32i)T + (-18.5 + 32.0i)T^{2}
41 1+2.55T+41T2 1 + 2.55T + 41T^{2}
43 1+2.73T+43T2 1 + 2.73T + 43T^{2}
47 1+(2.68+4.65i)T+(−23.5+40.7i)T2 1 + (2.68 + 4.65i)T + (-23.5 + 40.7i)T^{2}
53 1+(−4.89+8.47i)T+(−26.5−45.8i)T2 1 + (-4.89 + 8.47i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.25−2.16i)T+(−29.5−51.0i)T2 1 + (1.25 - 2.16i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.45−9.45i)T+(−30.5+52.8i)T2 1 + (-5.45 - 9.45i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.16−3.74i)T+(−33.5−58.0i)T2 1 + (2.16 - 3.74i)T + (-33.5 - 58.0i)T^{2}
71 1−10.6T+71T2 1 - 10.6T + 71T^{2}
73 1+(−2.58+4.48i)T+(−36.5−63.2i)T2 1 + (-2.58 + 4.48i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.271−0.469i)T+(−39.5+68.4i)T2 1 + (-0.271 - 0.469i)T + (-39.5 + 68.4i)T^{2}
83 1−15.2T+83T2 1 - 15.2T + 83T^{2}
89 1+(4.61+7.99i)T+(−44.5+77.0i)T2 1 + (4.61 + 7.99i)T + (-44.5 + 77.0i)T^{2}
97 1+1.26T+97T2 1 + 1.26T + 97T^{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

−10.59687253695894961458499949695, −9.775845539081196593750151760289, −9.076713134543225759603710745685, −8.023343557651541048360356550510, −6.78621408059178065761648644496, −5.56558715453895577651576444654, −5.26596210631090284479308473197, −4.41297229559688395789238665096, −3.61606488902854498404339955160, −0.71832641340275595539265413317, 1.41336570396533091983992863286, 2.58697800231998390392696015595, 3.58802049802836212621727640326, 5.06092016791022582861657263712, 6.40514107658929293048226730825, 6.85146845471497203510902445510, 7.71881962462945203706729332311, 8.258895861851563418286777164625, 10.38592953046474334582590630908, 10.90498238355645624648308928172

Graph of the ZZ-function along the critical line