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Properties

Label 2-637-13.10-c1-0-13
Degree $2$
Conductor $637$
Sign $-0.407 - 0.913i$
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  + (0.489 − 0.282i)2-s + (1.54 + 2.67i)3-s + (−0.840 + 1.45i)4-s − 1.80i·5-s + (1.51 + 0.874i)6-s + 2.08i·8-s + (−3.27 + 5.67i)9-s + (−0.510 − 0.884i)10-s + (3.10 − 1.79i)11-s − 5.19·12-s + (−0.328 + 3.59i)13-s + (4.83 − 2.78i)15-s + (−1.09 − 1.89i)16-s + (−3.07 + 5.32i)17-s + 3.70i·18-s + (−2.17 − 1.25i)19-s + ⋯
L(s)  = 1  + (0.346 − 0.199i)2-s + (0.892 + 1.54i)3-s + (−0.420 + 0.727i)4-s − 0.807i·5-s + (0.618 + 0.356i)6-s + 0.735i·8-s + (−1.09 + 1.89i)9-s + (−0.161 − 0.279i)10-s + (0.935 − 0.540i)11-s − 1.49·12-s + (−0.0909 + 0.995i)13-s + (1.24 − 0.720i)15-s + (−0.272 − 0.472i)16-s + (−0.745 + 1.29i)17-s + 0.873i·18-s + (−0.498 − 0.288i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.407 - 0.913i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.407 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10930 + 1.70965i\)
\(L(\frac12)\) \(\approx\) \(1.10930 + 1.70965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.328 - 3.59i)T \)
good2 \( 1 + (-0.489 + 0.282i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.54 - 2.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.80iT - 5T^{2} \)
11 \( 1 + (-3.10 + 1.79i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.07 - 5.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.17 + 1.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.06 - 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.55 + 4.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.55iT - 31T^{2} \)
37 \( 1 + (-10.1 + 5.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.61 + 0.933i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.711 - 1.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.86iT - 47T^{2} \)
53 \( 1 - 2.02T + 53T^{2} \)
59 \( 1 + (7.88 + 4.55i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.73 + 4.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.07 + 3.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.89 - 2.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.35iT - 73T^{2} \)
79 \( 1 - 5.20T + 79T^{2} \)
83 \( 1 + 9.42iT - 83T^{2} \)
89 \( 1 + (-0.307 + 0.177i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 - 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.03664378385960904969015479322, −9.563376429202851573116000061711, −9.145864202142006402743321143028, −8.603450878824802212502434064942, −7.79149269426210730323477530490, −6.09420252728099199601554803320, −4.77145611808100631106000689653, −4.18940309774041184856870451489, −3.64510392128082030147818327385, −2.28542708321650396383540543635, 0.955771462717856136297888983713, 2.32765997752677557523522029529, 3.35399092702755160004227323984, 4.79522929959880259177626283794, 6.17045997937113417572030868481, 6.80809458503353423892392495472, 7.33800735526813666682950400543, 8.551134007880860712680599014745, 9.225149293702473278255345112834, 10.22851108376922689341793634497

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