Properties

Label 2-637-91.80-c1-0-2
Degree $2$
Conductor $637$
Sign $0.135 + 0.990i$
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  + (0.664 + 2.47i)2-s + 2.69i·3-s + (−3.97 + 2.29i)4-s + (0.103 − 0.384i)5-s + (−6.68 + 1.79i)6-s + (−4.70 − 4.70i)8-s − 4.27·9-s + 1.02·10-s + (2.56 + 2.56i)11-s + (−6.19 − 10.7i)12-s + (−3.44 − 1.06i)13-s + (1.03 + 0.277i)15-s + (3.95 − 6.84i)16-s + (−2.04 − 3.54i)17-s + (−2.83 − 10.5i)18-s + (0.569 + 0.569i)19-s + ⋯
L(s)  = 1  + (0.469 + 1.75i)2-s + 1.55i·3-s + (−1.98 + 1.14i)4-s + (0.0461 − 0.172i)5-s + (−2.72 + 0.731i)6-s + (−1.66 − 1.66i)8-s − 1.42·9-s + 0.323·10-s + (0.774 + 0.774i)11-s + (−1.78 − 3.09i)12-s + (−0.955 − 0.294i)13-s + (0.267 + 0.0717i)15-s + (0.987 − 1.71i)16-s + (−0.496 − 0.860i)17-s + (−0.668 − 2.49i)18-s + (0.130 + 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.135 + 0.990i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03262 - 0.901347i\)
\(L(\frac12)\) \(\approx\) \(1.03262 - 0.901347i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.44 + 1.06i)T \)
good2 \( 1 + (-0.664 - 2.47i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 2.69iT - 3T^{2} \)
5 \( 1 + (-0.103 + 0.384i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.56 - 2.56i)T + 11iT^{2} \)
17 \( 1 + (2.04 + 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.569 - 0.569i)T + 19iT^{2} \)
23 \( 1 + (-4.41 - 2.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.06 - 1.62i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.73 - 0.463i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.578 - 2.15i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.19 - 2.19i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.17 - 1.92i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 2.77iT - 61T^{2} \)
67 \( 1 + (-3.55 + 3.55i)T - 67iT^{2} \)
71 \( 1 + (0.582 + 2.17i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.28 - 4.78i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.80 - 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.36 + 5.36i)T + 83iT^{2} \)
89 \( 1 + (3.09 + 11.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.71 + 2.06i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.10885844003412383544548265185, −9.979351387838336375771745559464, −9.184379909166984199573927961296, −8.857581522077836264538653275312, −7.43770111743984461290165272430, −6.89085949184419067006516163015, −5.53616993113629560825141290201, −4.93160894142671197931751290093, −4.32940509811098068427145598738, −3.23509365226687853149762015643, 0.65901264409958786112845003667, 1.84690906328714553594185212929, 2.68444003557805354482941747810, 3.84732166593235055036799375147, 5.10008427159374015897416069117, 6.28640453658847923206046020400, 7.11225857107440327458960437390, 8.444991005706306537786295872824, 9.113424328371819586989315107190, 10.23747708379045968636091763805

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