Properties

Label 2-637-49.39-c1-0-44
Degree $2$
Conductor $637$
Sign $0.827 - 0.560i$
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  + (2.72 + 0.410i)2-s + (0.680 − 0.463i)3-s + (5.34 + 1.64i)4-s + (0.189 + 2.52i)5-s + (2.04 − 0.984i)6-s + (−1.91 − 1.82i)7-s + (8.92 + 4.29i)8-s + (−0.848 + 2.16i)9-s + (−0.522 + 6.96i)10-s + (−2.37 − 6.04i)11-s + (4.40 − 1.35i)12-s + (−0.623 + 0.781i)13-s + (−4.47 − 5.75i)14-s + (1.30 + 1.63i)15-s + (13.3 + 9.07i)16-s + (−3.06 − 2.84i)17-s + ⋯
L(s)  = 1  + (1.92 + 0.290i)2-s + (0.392 − 0.267i)3-s + (2.67 + 0.824i)4-s + (0.0847 + 1.13i)5-s + (0.834 − 0.401i)6-s + (−0.724 − 0.689i)7-s + (3.15 + 1.51i)8-s + (−0.282 + 0.720i)9-s + (−0.165 + 2.20i)10-s + (−0.714 − 1.82i)11-s + (1.27 − 0.391i)12-s + (−0.172 + 0.216i)13-s + (−1.19 − 1.53i)14-s + (0.336 + 0.421i)15-s + (3.32 + 2.26i)16-s + (−0.743 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.827 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.58604 + 1.40688i\)
\(L(\frac12)\) \(\approx\) \(4.58604 + 1.40688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.91 + 1.82i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-2.72 - 0.410i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-0.680 + 0.463i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.189 - 2.52i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (2.37 + 6.04i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (3.06 + 2.84i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (0.335 + 0.581i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.258 + 0.240i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.18 + 5.20i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (3.77 - 6.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.53 + 0.780i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (7.57 + 3.64i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.72 + 2.75i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-9.21 - 1.38i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (0.141 + 0.0435i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.485 + 6.48i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-0.758 + 0.234i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (4.66 - 8.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.58 - 11.3i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (12.3 - 1.85i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (2.18 + 3.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.55 - 10.7i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.924 - 2.35i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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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

−10.91091923103680330565006052499, −10.42716710971485633501339241138, −8.557638225555027481765918007569, −7.45817363142032949785067690608, −6.93281585864492040858316668512, −6.08383616059343262906879236563, −5.24529246577546412438230326815, −3.93739449928829693466302693572, −2.98314379701685450860505488118, −2.54261561538919337292404238185, 1.93027045814201924171357138322, 2.90242099447078192655728579915, 4.08572415515277244808071962623, 4.78756545327129116780197504205, 5.64909345885626705592575953887, 6.50919327371655708427266101704, 7.56132856376664180164854256659, 8.942118367684949657182352524963, 9.784267924537342082541685848763, 10.63931245040319295479742941305

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