Properties

Label 2-637-49.37-c1-0-42
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: $\chi_{637} (625, \cdot )$
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

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

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