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Properties

Label 2-637-91.25-c1-0-26
Degree $2$
Conductor $637$
Sign $0.671 + 0.740i$
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  + (−0.596 + 0.344i)2-s + (1.10 − 1.91i)3-s + (−0.762 + 1.32i)4-s + (−2.78 + 1.60i)5-s + 1.52i·6-s − 2.42i·8-s + (−0.951 − 1.64i)9-s + (1.10 − 1.91i)10-s + (−2.32 − 1.34i)11-s + (1.68 + 2.92i)12-s + (3.59 − 0.311i)13-s + 7.11i·15-s + (−0.688 − 1.19i)16-s + (1.79 − 3.11i)17-s + (1.13 + 0.655i)18-s + (7.40 − 4.27i)19-s + ⋯
L(s)  = 1  + (−0.421 + 0.243i)2-s + (0.639 − 1.10i)3-s + (−0.381 + 0.660i)4-s + (−1.24 + 0.718i)5-s + 0.622i·6-s − 0.858i·8-s + (−0.317 − 0.549i)9-s + (0.350 − 0.606i)10-s + (−0.702 − 0.405i)11-s + (0.487 + 0.844i)12-s + (0.996 − 0.0862i)13-s + 1.83i·15-s + (−0.172 − 0.298i)16-s + (0.435 − 0.754i)17-s + (0.267 + 0.154i)18-s + (1.69 − 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916220 - 0.405896i\)
\(L(\frac12)\) \(\approx\) \(0.916220 - 0.405896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.311i)T \)
good2 \( 1 + (0.596 - 0.344i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.10 + 1.91i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.78 - 1.60i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.32 + 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.79 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.40 + 4.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.64 + 2.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + (-5.05 - 2.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.40 + 1.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.755iT - 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 + (-1.63 + 0.941i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 - 2.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.34 + 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.52 + 7.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.371 - 0.214i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + (5.01 + 2.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.23 - 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (-4.64 + 2.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.62iT - 97T^{2} \)
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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

−10.48717288289563434223391349652, −9.252009326453654785772065611486, −8.304415518044086649691601854596, −7.85502103568820051543420258333, −7.28003416175838200289674127893, −6.51748295136747426315431110148, −4.79847805597651361839193434639, −3.35690707646734718484851093250, −2.92061728749964262414107840357, −0.71896712040728990386644392137, 1.23736049498745046627837396104, 3.24864006071389750476658841202, 4.12191888787219310259406136724, 4.90259976509776147195371761338, 5.90437553808698403824457396800, 7.75540572757409042009208453209, 8.262785633505028062822781118991, 9.009431255028228185651375927364, 9.922901848628245753655095012473, 10.30607719325196031801682716226

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