Properties

Label 2-637-91.88-c1-0-11
Degree $2$
Conductor $637$
Sign $0.823 + 0.566i$
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  + (−2.24 − 1.29i)2-s − 0.518·3-s + (2.35 + 4.07i)4-s + (−1.39 + 0.806i)5-s + (1.16 + 0.671i)6-s − 6.99i·8-s − 2.73·9-s + 4.17·10-s − 2.70i·11-s + (−1.21 − 2.11i)12-s + (−2.36 + 2.71i)13-s + (0.723 − 0.417i)15-s + (−4.34 + 7.53i)16-s + (−1.56 − 2.70i)17-s + (6.12 + 3.53i)18-s − 3.68i·19-s + ⋯
L(s)  = 1  + (−1.58 − 0.915i)2-s − 0.299·3-s + (1.17 + 2.03i)4-s + (−0.624 + 0.360i)5-s + (0.474 + 0.273i)6-s − 2.47i·8-s − 0.910·9-s + 1.31·10-s − 0.815i·11-s + (−0.351 − 0.609i)12-s + (−0.656 + 0.753i)13-s + (0.186 − 0.107i)15-s + (−1.08 + 1.88i)16-s + (−0.379 − 0.656i)17-s + (1.44 + 0.833i)18-s − 0.844i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.823 + 0.566i$
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: \((2,\ 637,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376839 - 0.117136i\)
\(L(\frac12)\) \(\approx\) \(0.376839 - 0.117136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 2.71i)T \)
good2 \( 1 + (2.24 + 1.29i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 0.518T + 3T^{2} \)
5 \( 1 + (1.39 - 0.806i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.70iT - 11T^{2} \)
17 \( 1 + (1.56 + 2.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.68iT - 19T^{2} \)
23 \( 1 + (-0.993 + 1.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.07 - 5.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.15 - 2.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.66 + 3.85i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 - 2.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.913 + 0.527i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.63 - 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.89 + 5.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.92T + 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 + (-1.17 - 0.675i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.88 + 4.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.69iT - 83T^{2} \)
89 \( 1 + (1.52 + 0.879i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.4 - 7.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.67526396999688094871362802329, −9.574756559730280560449987438758, −8.863900997069441388085856935565, −8.196619112792557235691085669220, −7.22869814973935348263237056006, −6.44654024930490653198588451613, −4.77945050151903595906820342868, −3.24062612781280193779829139892, −2.53464438834220998502540587092, −0.70366555032453540690329072153, 0.63969520050714243277928791517, 2.39305808741371959166309843252, 4.40137338271819921864584958456, 5.65425382077585683946275527844, 6.32603022152793095531206010852, 7.48433149041329000339765440355, 8.074335587175804391405070188803, 8.652128252743173736343428585761, 9.831077271988308185465732287935, 10.20296647062042582024204021306

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