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Properties

Label 2-637-91.9-c1-0-21
Degree $2$
Conductor $637$
Sign $-0.206 + 0.978i$
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  + (−0.115 − 0.200i)2-s − 3.32·3-s + (0.973 − 1.68i)4-s + (1.11 − 1.93i)5-s + (0.384 + 0.665i)6-s − 0.913·8-s + 8.03·9-s − 0.516·10-s + 3.32·11-s + (−3.23 + 5.60i)12-s + (3.40 + 1.19i)13-s + (−3.70 + 6.41i)15-s + (−1.84 − 3.18i)16-s + (0.687 − 1.19i)17-s + (−0.929 − 1.61i)18-s + 3.23·19-s + ⋯
L(s)  = 1  + (−0.0817 − 0.141i)2-s − 1.91·3-s + (0.486 − 0.842i)4-s + (0.498 − 0.864i)5-s + (0.156 + 0.271i)6-s − 0.322·8-s + 2.67·9-s − 0.163·10-s + 1.00·11-s + (−0.933 + 1.61i)12-s + (0.943 + 0.330i)13-s + (−0.957 + 1.65i)15-s + (−0.460 − 0.797i)16-s + (0.166 − 0.288i)17-s + (−0.219 − 0.379i)18-s + 0.742·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641362 - 0.790969i\)
\(L(\frac12)\) \(\approx\) \(0.641362 - 0.790969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 - 1.19i)T \)
good2 \( 1 + (0.115 + 0.200i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 + (-1.11 + 1.93i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 3.32T + 11T^{2} \)
17 \( 1 + (-0.687 + 1.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.23T + 19T^{2} \)
23 \( 1 + (0.419 + 0.726i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.303 + 0.525i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.857 + 1.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.776 + 1.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.58 + 7.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.615 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.814 + 1.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.19 + 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.41 - 7.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (-2.60 - 4.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.98 - 3.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.64T + 83T^{2} \)
89 \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.67 - 13.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49386699597442939157623020203, −9.678207500205579075655773469167, −9.002586909542671933465301034335, −7.23137542812400133985103925474, −6.45156920104138705447736964984, −5.73417361069565332695854616801, −5.18360362835443752393634240005, −4.09476038241205978186270806614, −1.61694690443527185499187755215, −0.859325852437997164520598120889, 1.43782393917450151760126225531, 3.27363242002891178033633061143, 4.40361974700282938056144982261, 5.78847926025362385670630855212, 6.34548370986875280828105553120, 6.91509094945207405817494087795, 7.86886815169259060388020470052, 9.305606373971137493536447471574, 10.30205627149380125484384122629, 11.02047091410827688603852082743

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