Properties

Label 2-637-49.4-c1-0-46
Degree $2$
Conductor $637$
Sign $0.426 - 0.904i$
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  + (1.41 + 0.964i)2-s + (1.37 + 1.27i)3-s + (0.340 + 0.867i)4-s + (3.26 − 1.00i)5-s + (0.715 + 3.13i)6-s + (−0.397 + 2.61i)7-s + (0.406 − 1.78i)8-s + (0.0391 + 0.522i)9-s + (5.59 + 1.72i)10-s + (0.0906 − 1.21i)11-s + (−0.639 + 1.62i)12-s + (−0.900 + 0.433i)13-s + (−3.08 + 3.31i)14-s + (5.78 + 2.78i)15-s + (3.66 − 3.39i)16-s + (−6.85 + 1.03i)17-s + ⋯
L(s)  = 1  + (1.00 + 0.682i)2-s + (0.794 + 0.737i)3-s + (0.170 + 0.433i)4-s + (1.46 − 0.450i)5-s + (0.292 + 1.27i)6-s + (−0.150 + 0.988i)7-s + (0.143 − 0.630i)8-s + (0.0130 + 0.174i)9-s + (1.77 + 0.546i)10-s + (0.0273 − 0.364i)11-s + (−0.184 + 0.470i)12-s + (−0.249 + 0.120i)13-s + (−0.824 + 0.886i)14-s + (1.49 + 0.719i)15-s + (0.915 − 0.849i)16-s + (−1.66 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07426 + 1.95007i\)
\(L(\frac12)\) \(\approx\) \(3.07426 + 1.95007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.397 - 2.61i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
good2 \( 1 + (-1.41 - 0.964i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (-1.37 - 1.27i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-3.26 + 1.00i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.0906 + 1.21i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (6.85 - 1.03i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (1.96 + 3.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.83 + 0.880i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-3.59 - 4.50i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (1.66 - 2.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.51 - 6.41i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (2.12 - 9.30i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.0222 - 0.0974i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.39 + 3.67i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (1.27 + 3.25i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-8.95 - 2.76i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (2.52 - 6.42i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-0.369 + 0.640i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.90 + 8.66i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.731 + 0.498i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (5.42 + 9.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.92 - 3.33i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.337 - 4.49i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−10.38464286700618919562694485069, −9.721649269851794104544820053029, −8.962594198606488884616322565238, −8.469912364065925512490372742703, −6.55808506162542855605240712764, −6.26073501391504120701403621073, −5.11252174331924259497640561786, −4.53940856340012921047166725595, −3.18916005742255424931930640546, −2.07795655620719670495654177111, 2.04349715444666493883276095219, 2.22927888782960163550147810977, 3.62026285223987386591009795245, 4.63976866577730632572044407434, 5.86772383408206481010490541358, 6.76079367144483390184280789122, 7.68920222937156397208649772534, 8.670663663525342413825518416612, 9.825702708739742528211844198525, 10.48436497205212707235156175916

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