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Properties

Label 2-637-91.81-c1-0-12
Degree $2$
Conductor $637$
Sign $-0.113 - 0.993i$
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  + (−0.207 + 0.358i)2-s + 1.41·3-s + (0.914 + 1.58i)4-s + (0.914 + 1.58i)5-s + (−0.292 + 0.507i)6-s − 1.58·8-s − 0.999·9-s − 0.757·10-s + 0.585·11-s + (1.29 + 2.23i)12-s + (−3.5 + 0.866i)13-s + (1.29 + 2.23i)15-s + (−1.49 + 2.59i)16-s + (2.91 + 5.04i)17-s + (0.207 − 0.358i)18-s + 6·19-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + 0.816·3-s + (0.457 + 0.791i)4-s + (0.408 + 0.708i)5-s + (−0.119 + 0.207i)6-s − 0.560·8-s − 0.333·9-s − 0.239·10-s + 0.176·11-s + (0.373 + 0.646i)12-s + (−0.970 + 0.240i)13-s + (0.333 + 0.578i)15-s + (−0.374 + 0.649i)16-s + (0.706 + 1.22i)17-s + (0.0488 − 0.0845i)18-s + 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.113 - 0.993i$
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: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29379 + 1.45027i\)
\(L(\frac12)\) \(\approx\) \(1.29379 + 1.45027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + (-0.914 - 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.585T + 11T^{2} \)
17 \( 1 + (-2.91 - 5.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.08 + 3.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.29 - 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.74 + 8.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0857 + 0.148i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.70 - 2.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.82 - 3.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.17T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + (-5.82 + 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.32 + 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.878 + 1.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + (7.65 - 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.41 + 9.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.77646537097441521174304311614, −9.755983214312579403956248465586, −9.030688217691170122302852133209, −8.006391139298440067623832749936, −7.50030395644895715263044612988, −6.52821998022063435175605814625, −5.57868776707185767015735344749, −3.89654188030203016944750728332, −2.99561933797327250604685367889, −2.19084714748773218598367160122, 1.03895594307536833891020496875, 2.39559176952680047883823260014, 3.28979579353599899762531540391, 5.15256136406514950223557928332, 5.46634208706761620084945069477, 6.93023891381082877706877621861, 7.75369016668313674720816445395, 8.884795186641783424662785357939, 9.620863100779677879620536808996, 9.909314006762945342238332768600

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