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Properties

Label 2-637-91.25-c1-0-32
Degree $2$
Conductor $637$
Sign $0.0695 + 0.997i$
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  + (−1.97 + 1.14i)2-s + (1.57 − 2.72i)3-s + (1.61 − 2.78i)4-s + (1.84 − 1.06i)5-s + 7.19i·6-s + 2.78i·8-s + (−3.46 − 5.99i)9-s + (−2.42 + 4.20i)10-s + (−0.267 − 0.154i)11-s + (−5.07 − 8.78i)12-s + (3.22 − 1.62i)13-s − 6.69i·15-s + (0.0349 + 0.0605i)16-s + (0.887 − 1.53i)17-s + (13.7 + 7.91i)18-s + (1.54 − 0.890i)19-s + ⋯
L(s)  = 1  + (−1.39 + 0.807i)2-s + (0.909 − 1.57i)3-s + (0.805 − 1.39i)4-s + (0.823 − 0.475i)5-s + 2.93i·6-s + 0.985i·8-s + (−1.15 − 1.99i)9-s + (−0.767 + 1.32i)10-s + (−0.0805 − 0.0465i)11-s + (−1.46 − 2.53i)12-s + (0.893 − 0.449i)13-s − 1.72i·15-s + (0.00874 + 0.0151i)16-s + (0.215 − 0.372i)17-s + (3.22 + 1.86i)18-s + (0.353 − 0.204i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0695 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0695 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0695 + 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0695 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833283 - 0.777214i\)
\(L(\frac12)\) \(\approx\) \(0.833283 - 0.777214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 + 1.62i)T \)
good2 \( 1 + (1.97 - 1.14i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.57 + 2.72i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.84 + 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.267 + 0.154i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.887 + 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.54 + 0.890i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.575 - 0.996i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 + (-3.98 - 2.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.79 - 2.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.72iT - 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (8.24 - 4.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.72 - 6.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.03 + 4.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.72 + 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 + (-10.2 - 5.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.96 - 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + (-1.43 + 0.829i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.66iT - 97T^{2} \)
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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

−9.825844179286023883107616810761, −9.183704656747390804948174186679, −8.436606234710488404866635040994, −7.951408982265326259767674575963, −7.02038739705871656610396436207, −6.37268713931310413886553521399, −5.48830903591705295794762794573, −3.17372564069438995671305097016, −1.76585602550042000802632089459, −0.942791724917145975217714649004, 1.86666090065912908482474729187, 2.88901412124858777108942404710, 3.74728514012643936522398735078, 5.08295775832781182261761167639, 6.43144883094742785090352377223, 7.991773513721726410149120958817, 8.488225737586119635428862612862, 9.444751198764305874680385994108, 9.754800346015800367714716030258, 10.54338004018816983990250507446

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