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Properties

Label 2-637-7.4-c1-0-10
Degree $2$
Conductor $637$
Sign $-0.947 + 0.318i$
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  + (1.22 + 2.11i)2-s + (−0.333 + 0.578i)3-s + (−1.99 + 3.45i)4-s + (0.455 + 0.788i)5-s − 1.63·6-s − 4.87·8-s + (1.27 + 2.21i)9-s + (−1.11 + 1.92i)10-s + (−1.83 + 3.18i)11-s + (−1.33 − 2.30i)12-s − 13-s − 0.607·15-s + (−1.97 − 3.42i)16-s + (3.59 − 6.22i)17-s + (−3.12 + 5.41i)18-s + (−0.989 − 1.71i)19-s + ⋯
L(s)  = 1  + (0.865 + 1.49i)2-s + (−0.192 + 0.333i)3-s + (−0.997 + 1.72i)4-s + (0.203 + 0.352i)5-s − 0.667·6-s − 1.72·8-s + (0.425 + 0.737i)9-s + (−0.352 + 0.610i)10-s + (−0.554 + 0.960i)11-s + (−0.384 − 0.666i)12-s − 0.277·13-s − 0.156·15-s + (−0.493 − 0.855i)16-s + (0.871 − 1.50i)17-s + (−0.736 + 1.27i)18-s + (−0.226 − 0.392i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323993 - 1.98105i\)
\(L(\frac12)\) \(\approx\) \(0.323993 - 1.98105i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (-1.22 - 2.11i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.333 - 0.578i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.455 - 0.788i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.83 - 3.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.59 + 6.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.989 + 1.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.298 - 0.516i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.64T + 29T^{2} \)
31 \( 1 + (3.54 - 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.355 + 0.615i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (-6.05 - 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.72 + 9.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.79 + 8.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.49 - 6.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.614 - 1.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + (-3.26 + 5.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.76 - 9.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.16T + 83T^{2} \)
89 \( 1 + (-6.42 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.09T + 97T^{2} \)
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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

−10.99771184634421125170585705967, −10.10241487302533065992884985546, −9.225675585604194263249880332712, −7.940106364815942595924173342021, −7.31466933166626526675765890112, −6.71579981810107386474730949479, −5.31012588225048534054257683097, −5.06868315370243369766181675224, −3.99414902810521792244796202135, −2.55212803839924667043143412570, 0.889740653057662034786035416057, 2.04800182541080329116087102410, 3.43137868067861434165354849843, 4.08859780134221568310393956222, 5.50625150631879163333383535671, 5.91149810246871844407854638259, 7.42144056693408264561366562974, 8.643557519210928867061624461225, 9.590253632991010591466616603021, 10.40394733412138908552717050890

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