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Properties

Label 2-637-13.10-c1-0-1
Degree $2$
Conductor $637$
Sign $0.454 - 0.890i$
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  + (−2.04 + 1.18i)2-s + (−1.49 − 2.58i)3-s + (1.79 − 3.10i)4-s − 3.20i·5-s + (6.11 + 3.53i)6-s + 3.75i·8-s + (−2.96 + 5.12i)9-s + (3.78 + 6.55i)10-s + (−3.19 + 1.84i)11-s − 10.7·12-s + (−2.99 + 2.00i)13-s + (−8.28 + 4.78i)15-s + (−0.848 − 1.47i)16-s + (−0.733 + 1.27i)17-s − 13.9i·18-s + (5.11 + 2.95i)19-s + ⋯
L(s)  = 1  + (−1.44 + 0.835i)2-s + (−0.862 − 1.49i)3-s + (0.896 − 1.55i)4-s − 1.43i·5-s + (2.49 + 1.44i)6-s + 1.32i·8-s + (−0.986 + 1.70i)9-s + (1.19 + 2.07i)10-s + (−0.964 + 0.556i)11-s − 3.09·12-s + (−0.831 + 0.555i)13-s + (−2.13 + 1.23i)15-s + (−0.212 − 0.367i)16-s + (−0.177 + 0.308i)17-s − 3.29i·18-s + (1.17 + 0.678i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.454 - 0.890i$
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: \((2,\ 637,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152455 + 0.0933970i\)
\(L(\frac12)\) \(\approx\) \(0.152455 + 0.0933970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.99 - 2.00i)T \)
good2 \( 1 + (2.04 - 1.18i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.49 + 2.58i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.20iT - 5T^{2} \)
11 \( 1 + (3.19 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.733 - 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.345 + 0.597i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.39 - 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.10iT - 31T^{2} \)
37 \( 1 + (-1.09 + 0.629i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.224 - 0.129i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.80 - 4.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 - 0.485T + 53T^{2} \)
59 \( 1 + (7.55 + 4.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.73 - 9.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.296 + 0.170i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.10 + 4.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.259iT - 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (2.16 - 1.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.58 + 2.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.50053447807864291929321798286, −9.647458843589213805022580828375, −8.731905146950390585750841749210, −7.88295689740976526742684567796, −7.46117224282430240815414021220, −6.56944870186858254076523834315, −5.58646217237193230349515218693, −4.90609260040474564992238480761, −1.97717945457879734583843236279, −1.03292839382281337804630234219, 0.21132604792190576579424526298, 2.76575418317015395241280060085, 3.22553520019744455856179371417, 4.83111701058372395484789642112, 5.86995552364562302130491817467, 7.14171437571966791650421453971, 7.972715387408169504984698382170, 9.239938258232414817812290586637, 9.855604730871858255133431999333, 10.44158399076949800065962428700

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