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Properties

Label 2-637-91.24-c1-0-10
Degree $2$
Conductor $637$
Sign $0.902 - 0.431i$
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  + (0.369 − 0.0990i)2-s − 0.914i·3-s + (−1.60 + 0.926i)4-s + (−3.58 − 0.959i)5-s + (−0.0906 − 0.338i)6-s + (−1.04 + 1.04i)8-s + 2.16·9-s − 1.41·10-s + (0.0619 − 0.0619i)11-s + (0.847 + 1.46i)12-s + (1.63 + 3.21i)13-s + (−0.877 + 3.27i)15-s + (1.57 − 2.72i)16-s + (2.94 + 5.10i)17-s + (0.799 − 0.214i)18-s + (2.62 − 2.62i)19-s + ⋯
L(s)  = 1  + (0.261 − 0.0700i)2-s − 0.528i·3-s + (−0.802 + 0.463i)4-s + (−1.60 − 0.429i)5-s + (−0.0369 − 0.138i)6-s + (−0.368 + 0.368i)8-s + 0.721·9-s − 0.448·10-s + (0.0186 − 0.0186i)11-s + (0.244 + 0.423i)12-s + (0.453 + 0.891i)13-s + (−0.226 + 0.845i)15-s + (0.392 − 0.680i)16-s + (0.714 + 1.23i)17-s + (0.188 − 0.0505i)18-s + (0.602 − 0.602i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988448 + 0.224240i\)
\(L(\frac12)\) \(\approx\) \(0.988448 + 0.224240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 3.21i)T \)
good2 \( 1 + (-0.369 + 0.0990i)T + (1.73 - i)T^{2} \)
3 \( 1 + 0.914iT - 3T^{2} \)
5 \( 1 + (3.58 + 0.959i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0619 + 0.0619i)T - 11iT^{2} \)
17 \( 1 + (-2.94 - 5.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.62 + 2.62i)T - 19iT^{2} \)
23 \( 1 + (0.386 + 0.223i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.706 - 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.520 - 1.94i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.686 - 2.56i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.00 - 0.804i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.64 - 4.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.36 - 8.84i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.28 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.85 + 6.90i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 0.127iT - 61T^{2} \)
67 \( 1 + (7.04 + 7.04i)T + 67iT^{2} \)
71 \( 1 + (9.83 - 2.63i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.84 + 2.37i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.75 - 3.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.17 - 2.17i)T - 83iT^{2} \)
89 \( 1 + (1.63 - 0.437i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.452 + 1.68i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.90787803954762409186626364273, −9.560282862906260871481053313764, −8.726359949038714794824233906267, −7.897818026724551743497254524978, −7.44898636670669016935034810275, −6.20224139322364521910346616407, −4.68912714474452689156986741634, −4.17210716428537827725400937762, −3.24355398348274051591507901134, −1.13241447039591977514822520784, 0.68453764095211805426626169827, 3.28880816809449788037438330355, 3.91727511856803134496925311359, 4.80416708525346723997606616822, 5.74804464877924030311330971371, 7.17525776972637629398345289515, 7.80804369730115124844744464190, 8.795247007247620027909375012771, 9.819709186333658399673958755464, 10.40102568373665228471130633257

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