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Properties

Label 2-637-91.20-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.972 + 0.233i$
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  + (1.19 − 0.320i)2-s + (−1.92 − 1.11i)3-s + (−0.401 + 0.231i)4-s + (1.84 − 1.84i)5-s + (−2.66 − 0.713i)6-s + (−2.15 + 2.15i)8-s + (0.975 + 1.69i)9-s + (1.61 − 2.79i)10-s + (−0.365 − 1.36i)11-s + 1.03·12-s + (0.445 − 3.57i)13-s + (−5.60 + 1.50i)15-s + (−1.42 + 2.47i)16-s + (−1.41 − 2.44i)17-s + (1.71 + 1.71i)18-s + (−6.04 − 1.61i)19-s + ⋯
L(s)  = 1  + (0.846 − 0.226i)2-s + (−1.11 − 0.642i)3-s + (−0.200 + 0.115i)4-s + (0.823 − 0.823i)5-s + (−1.08 − 0.291i)6-s + (−0.763 + 0.763i)8-s + (0.325 + 0.563i)9-s + (0.510 − 0.884i)10-s + (−0.110 − 0.411i)11-s + 0.297·12-s + (0.123 − 0.992i)13-s + (−1.44 + 0.387i)15-s + (−0.357 + 0.618i)16-s + (−0.342 − 0.594i)17-s + (0.403 + 0.403i)18-s + (−1.38 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.972 + 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107237 - 0.906630i\)
\(L(\frac12)\) \(\approx\) \(0.107237 - 0.906630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.445 + 3.57i)T \)
good2 \( 1 + (-1.19 + 0.320i)T + (1.73 - i)T^{2} \)
3 \( 1 + (1.92 + 1.11i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.84 + 1.84i)T - 5iT^{2} \)
11 \( 1 + (0.365 + 1.36i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.41 + 2.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.04 + 1.61i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.882 + 0.509i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.66 - 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.72 - 4.72i)T - 31iT^{2} \)
37 \( 1 + (2.38 + 8.89i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.51 + 9.40i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.850 + 0.490i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.62 - 1.62i)T + 47iT^{2} \)
53 \( 1 - 5.05T + 53T^{2} \)
59 \( 1 + (1.97 - 7.36i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.75 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.6 + 3.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.97 + 11.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.17 + 6.17i)T + 73iT^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (0.445 - 0.445i)T - 83iT^{2} \)
89 \( 1 + (0.147 - 0.0396i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.87 + 0.771i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.59053226655951228579051571864, −9.102123472719239808249889287155, −8.658849963442570953467747750552, −7.26789230146078827922049727417, −6.14988863266448797502410231131, −5.44533834391175849897566932200, −5.00084324113962244338387904846, −3.61748981368974564735853561854, −2.09877634375847049450227884144, −0.41456243394925835027352582916, 2.18382536170560931224281188434, 3.90163874335656650103623733756, 4.56673803962891302918008755809, 5.60280498644248786416860828395, 6.26590703820603453155803230381, 6.77049492337750966831966547876, 8.462116419127305559121266539269, 9.770269756506626627202357829328, 10.02843314599648456632189377270, 11.02581801535497570117994620153

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