Properties

Label 2-637-91.81-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.803 + 0.595i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.330i)2-s − 0.381·3-s + (0.927 + 1.60i)4-s + (−0.190 − 0.330i)5-s + (0.0729 − 0.126i)6-s − 1.47·8-s − 2.85·9-s + 0.145·10-s − 4.85·11-s + (−0.354 − 0.613i)12-s + (−2.5 − 2.59i)13-s + (0.0729 + 0.126i)15-s + (−1.57 + 2.72i)16-s + (−3.73 − 6.47i)17-s + (0.545 − 0.944i)18-s + 4.85·19-s + ⋯
L(s)  = 1  + (−0.135 + 0.233i)2-s − 0.220·3-s + (0.463 + 0.802i)4-s + (−0.0854 − 0.147i)5-s + (0.0297 − 0.0515i)6-s − 0.520·8-s − 0.951·9-s + 0.0461·10-s − 1.46·11-s + (−0.102 − 0.177i)12-s + (−0.693 − 0.720i)13-s + (0.0188 + 0.0326i)15-s + (−0.393 + 0.681i)16-s + (−0.906 − 1.56i)17-s + (0.128 − 0.222i)18-s + 1.11·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (0.190 - 0.330i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.381T + 3T^{2} \)
5 \( 1 + (0.190 + 0.330i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.85T + 11T^{2} \)
17 \( 1 + (3.73 + 6.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.04 - 3.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.35 - 7.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.61 + 4.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.78 + 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.11 + 1.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.11 - 7.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 0.708T + 67T^{2} \)
71 \( 1 + (4.09 - 7.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (8.04 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.07 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.38007379119849548121637713621, −9.175503516444045614660392955669, −8.380001488722985264309081015692, −7.54322386340171729115201190047, −6.93190067328791730404683455113, −5.53756554414450238262609527407, −4.95170586053693634784177388736, −3.16748509140987734906855591612, −2.59550403565865417696718474522, 0, 2.00527308999423146027154061936, 2.94912396629309653339228610913, 4.60954723869384635520296905504, 5.62005578126017163805885598093, 6.27242090166650313935153716975, 7.38729008597001431775717312410, 8.333604882163238787259929773251, 9.385204885034420512337350535831, 10.18714678372792025162868558901

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