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Properties

Label 2-637-91.88-c1-0-25
Degree $2$
Conductor $637$
Sign $-0.369 + 0.929i$
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  + (−1.10 − 0.638i)2-s + 1.16·3-s + (−0.185 − 0.320i)4-s + (1.57 − 0.907i)5-s + (−1.29 − 0.745i)6-s + 3.02i·8-s − 1.63·9-s − 2.31·10-s − 2.77i·11-s + (−0.216 − 0.374i)12-s + (3.58 + 0.402i)13-s + (1.83 − 1.05i)15-s + (1.56 − 2.70i)16-s + (−1.37 − 2.37i)17-s + (1.80 + 1.04i)18-s − 5.86i·19-s + ⋯
L(s)  = 1  + (−0.781 − 0.451i)2-s + 0.674·3-s + (−0.0925 − 0.160i)4-s + (0.702 − 0.405i)5-s + (−0.527 − 0.304i)6-s + 1.06i·8-s − 0.545·9-s − 0.732·10-s − 0.837i·11-s + (−0.0624 − 0.108i)12-s + (0.993 + 0.111i)13-s + (0.473 − 0.273i)15-s + (0.390 − 0.675i)16-s + (−0.332 − 0.576i)17-s + (0.426 + 0.246i)18-s − 1.34i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.369 + 0.929i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.369 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.654782 - 0.964628i\)
\(L(\frac12)\) \(\approx\) \(0.654782 - 0.964628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 0.402i)T \)
good2 \( 1 + (1.10 + 0.638i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 1.16T + 3T^{2} \)
5 \( 1 + (-1.57 + 0.907i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.77iT - 11T^{2} \)
17 \( 1 + (1.37 + 2.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.86iT - 19T^{2} \)
23 \( 1 + (-3.49 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 - 1.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.50 + 0.871i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.51 - 3.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.76 - 3.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.24 + 9.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.66 + 1.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.08T + 61T^{2} \)
67 \( 1 - 5.01iT - 67T^{2} \)
71 \( 1 + (-2.35 - 1.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.64 + 3.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.97iT - 83T^{2} \)
89 \( 1 + (13.9 + 8.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.3 - 7.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.20230958368914958475715445450, −9.139133547080751585810664211293, −8.832167902799017860815643284480, −8.281512917923582006479428451159, −6.78548345454339551407738949312, −5.70465362351574176948036818808, −4.85716260644291384681765240731, −3.19319386348208341195562944068, −2.20469951334301012579259954273, −0.789959773270974674916534603992, 1.71252785414120807109111640745, 3.13270437050395411912452167684, 4.09137721298584513464914354457, 5.74750024585559983291568679830, 6.52050186999574875437345203457, 7.61698173585256232617047085475, 8.260885340067938784965460696891, 9.023310770213356103773698614934, 9.787988793663170026648509836876, 10.42745060257653637688993909709

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