Properties

Label 2-637-49.4-c1-0-13
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  + (−2.27 − 1.54i)2-s + (2.32 + 2.16i)3-s + (2.03 + 5.18i)4-s + (1.52 − 0.469i)5-s + (−1.94 − 8.52i)6-s + (−2.61 − 0.397i)7-s + (2.18 − 9.57i)8-s + (0.529 + 7.07i)9-s + (−4.19 − 1.29i)10-s + (−0.368 + 4.92i)11-s + (−6.46 + 16.4i)12-s + (0.900 − 0.433i)13-s + (5.33 + 4.95i)14-s + (4.56 + 2.19i)15-s + (−11.6 + 10.7i)16-s + (0.173 − 0.0261i)17-s + ⋯
L(s)  = 1  + (−1.60 − 1.09i)2-s + (1.34 + 1.24i)3-s + (1.01 + 2.59i)4-s + (0.681 − 0.210i)5-s + (−0.793 − 3.47i)6-s + (−0.988 − 0.150i)7-s + (0.772 − 3.38i)8-s + (0.176 + 2.35i)9-s + (−1.32 − 0.408i)10-s + (−0.111 + 1.48i)11-s + (−1.86 + 4.75i)12-s + (0.249 − 0.120i)13-s + (1.42 + 1.32i)14-s + (1.17 + 0.567i)15-s + (−2.90 + 2.69i)16-s + (0.0421 − 0.00635i)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

\(L(1)\) \(\approx\) \(0.855249 + 0.542484i\)
\(L(\frac12)\) \(\approx\) \(0.855249 + 0.542484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.61 + 0.397i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (2.27 + 1.54i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (-2.32 - 2.16i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-1.52 + 0.469i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.368 - 4.92i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (-0.173 + 0.0261i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.907 + 1.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.03 - 0.456i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-2.94 - 3.68i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.572 - 1.45i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (2.68 - 11.7i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.07 + 9.07i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-1.58 - 1.08i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (1.61 + 4.12i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (8.14 + 2.51i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (0.0565 - 0.144i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (1.89 - 3.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.83 + 7.31i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.801 + 0.546i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-1.33 - 2.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 5.34i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.386 - 5.15i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−10.23596528139740502805056824368, −9.728651215421307964620537882488, −9.376031749958311641265256794727, −8.634993309427398388162177525402, −7.74359804362126680638623127109, −6.80301392671935752540623968628, −4.68530433542093165675706802673, −3.58138075690125478898406614088, −2.78396365896075760094574602523, −1.84518993816402898343540408529, 0.78297045968095820870570016234, 2.08542871426559438644796849780, 3.12890018456920665850872907084, 5.87983826559426254575033925254, 6.29509111120811203900469033131, 7.04216616500451120202884015106, 7.951990179784276959613690789369, 8.629519859107698105758906383183, 9.139401147053214048748564658758, 9.913409676269947475609283923008

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