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Properties

Label 2-637-91.74-c1-0-36
Degree $2$
Conductor $637$
Sign $-0.775 - 0.630i$
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  − 0.268·2-s + (−0.571 − 0.989i)3-s − 1.92·4-s + (−1.28 − 2.21i)5-s + (0.153 + 0.265i)6-s + 1.05·8-s + (0.846 − 1.46i)9-s + (0.343 + 0.594i)10-s + (−1.97 − 3.41i)11-s + (1.10 + 1.90i)12-s + (3.15 − 1.74i)13-s + (−1.46 + 2.53i)15-s + 3.57·16-s − 0.785·17-s + (−0.227 + 0.393i)18-s + (−3.74 + 6.49i)19-s + ⋯
L(s)  = 1  − 0.189·2-s + (−0.329 − 0.571i)3-s − 0.964·4-s + (−0.572 − 0.992i)5-s + (0.0625 + 0.108i)6-s + 0.372·8-s + (0.282 − 0.488i)9-s + (0.108 + 0.188i)10-s + (−0.594 − 1.03i)11-s + (0.318 + 0.550i)12-s + (0.874 − 0.484i)13-s + (−0.378 + 0.654i)15-s + 0.893·16-s − 0.190·17-s + (−0.0535 + 0.0926i)18-s + (−0.859 + 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.775 - 0.630i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.775 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0978010 + 0.275314i\)
\(L(\frac12)\) \(\approx\) \(0.0978010 + 0.275314i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 + 1.74i)T \)
good2 \( 1 + 0.268T + 2T^{2} \)
3 \( 1 + (0.571 + 0.989i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.785T + 17T^{2} \)
19 \( 1 + (3.74 - 6.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 + (1.17 - 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.27 - 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 + (1.21 - 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.96T + 59T^{2} \)
61 \( 1 + (-4.72 + 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.15 + 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.384 + 0.665i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + 7.66T + 89T^{2} \)
97 \( 1 + (1.18 + 2.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00272051415152281057853512270, −8.981496495401771434482129655052, −8.164639478978027026586624355034, −7.906533019879858450314915470685, −6.20634531571228856546025689675, −5.61032337731391614556864928603, −4.32190628961618544281521080064, −3.61122334371031476969491097437, −1.34070592054714039825533346688, −0.19818429921661631477491650500, 2.28492842216554993704442385648, 3.96147616445803868709991449182, 4.39388880766029582957315598823, 5.51791292988569004383339090827, 6.78831902089783715995442368013, 7.65788722580451563031362068630, 8.499626959239745037282361888595, 9.598381172149713698328170512157, 10.22056400633139114828714574692, 10.95433698856714718376441868143

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