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Properties

Label 2-637-91.54-c1-0-17
Degree $2$
Conductor $637$
Sign $-0.451 + 0.892i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0825 − 0.0825i)2-s + (−2.25 − 1.29i)3-s + 1.98i·4-s + (−1.70 + 0.456i)5-s + (−0.293 + 0.0785i)6-s + (0.329 + 0.329i)8-s + (1.87 + 3.25i)9-s + (−0.103 + 0.178i)10-s + (1.89 − 0.506i)11-s + (2.58 − 4.47i)12-s + (1.85 + 3.09i)13-s + (4.43 + 1.18i)15-s − 3.91·16-s − 4.27·17-s + (0.423 + 0.113i)18-s + (1.10 − 4.12i)19-s + ⋯
L(s)  = 1  + (0.0583 − 0.0583i)2-s + (−1.29 − 0.750i)3-s + 0.993i·4-s + (−0.762 + 0.204i)5-s + (−0.119 + 0.0320i)6-s + (0.116 + 0.116i)8-s + (0.625 + 1.08i)9-s + (−0.0325 + 0.0564i)10-s + (0.570 − 0.152i)11-s + (0.745 − 1.29i)12-s + (0.515 + 0.857i)13-s + (1.14 + 0.306i)15-s − 0.979·16-s − 1.03·17-s + (0.0997 + 0.0267i)18-s + (0.253 − 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204244 - 0.332237i\)
\(L(\frac12)\) \(\approx\) \(0.204244 - 0.332237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1.85 - 3.09i)T \)
good2 \( 1 + (-0.0825 + 0.0825i)T - 2iT^{2} \)
3 \( 1 + (2.25 + 1.29i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.70 - 0.456i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.89 + 0.506i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 + (-1.10 + 4.12i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.39iT - 23T^{2} \)
29 \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.10 + 4.13i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.00 + 2.00i)T + 37iT^{2} \)
41 \( 1 + (-2.94 + 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.71 - 6.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.13 + 3.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.19 + 1.19i)T - 59iT^{2} \)
61 \( 1 + (-2.66 + 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.00510 - 0.0190i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.954 - 0.255i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.96 + 5.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.87 + 9.87i)T + 83iT^{2} \)
89 \( 1 + (5.68 - 5.68i)T - 89iT^{2} \)
97 \( 1 + (14.2 - 3.82i)T + (84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90479332367519476441647031550, −9.256933479145554993638506491570, −8.429424511511212521189682802605, −7.34800496496812920555209134679, −6.83398453642363304505605364565, −6.02246515085990435042817615505, −4.56873291885805171570181724707, −3.85315468450285957412962497633, −2.22882618532669364091370154725, −0.26963650482172148323046809431, 1.29825140219037848887732620286, 3.64596333818657275345147048770, 4.57665773711732733942581098342, 5.41437620965384616340465516102, 6.09242047140231035744516935942, 7.04926419149927532618378009806, 8.329132155619530814168342968958, 9.446624012125737354800206269729, 10.14863814427928669215826534270, 10.96134280338832096783120581021

Graph of the $Z$-function along the critical line