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Properties

Label 2-637-91.76-c1-0-1
Degree $2$
Conductor $637$
Sign $0.827 + 0.562i$
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  + (−0.573 + 2.14i)2-s + (−0.959 + 0.554i)3-s + (−2.52 − 1.45i)4-s + (−2.14 + 2.14i)5-s + (−0.635 − 2.37i)6-s + (1.43 − 1.43i)8-s + (−0.885 + 1.53i)9-s + (−3.36 − 5.82i)10-s + (0.705 + 0.188i)11-s + 3.22·12-s + (−2.65 + 2.44i)13-s + (0.869 − 3.24i)15-s + (−0.668 − 1.15i)16-s + (−1.62 + 2.81i)17-s + (−2.77 − 2.77i)18-s + (−1.03 − 3.85i)19-s + ⋯
L(s)  = 1  + (−0.405 + 1.51i)2-s + (−0.554 + 0.319i)3-s + (−1.26 − 0.728i)4-s + (−0.959 + 0.959i)5-s + (−0.259 − 0.968i)6-s + (0.506 − 0.506i)8-s + (−0.295 + 0.511i)9-s + (−1.06 − 1.84i)10-s + (0.212 + 0.0569i)11-s + 0.932·12-s + (−0.735 + 0.678i)13-s + (0.224 − 0.838i)15-s + (−0.167 − 0.289i)16-s + (−0.393 + 0.681i)17-s + (−0.654 − 0.654i)18-s + (−0.237 − 0.885i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.827 + 0.562i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.827 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200758 - 0.0617659i\)
\(L(\frac12)\) \(\approx\) \(0.200758 - 0.0617659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.65 - 2.44i)T \)
good2 \( 1 + (0.573 - 2.14i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (0.959 - 0.554i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.14 - 2.14i)T - 5iT^{2} \)
11 \( 1 + (-0.705 - 0.188i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.62 - 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.03 + 3.85i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.16 + 2.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.78 - 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.92 - 2.92i)T - 31iT^{2} \)
37 \( 1 + (-9.52 - 2.55i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.46 - 0.660i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.73 + 3.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.24 - 1.24i)T + 47iT^{2} \)
53 \( 1 + 6.74T + 53T^{2} \)
59 \( 1 + (-3.18 + 0.853i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.30 + 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 6.27i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (8.06 - 2.16i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.45 - 4.45i)T + 73iT^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (-10.4 + 10.4i)T - 83iT^{2} \)
89 \( 1 + (2.42 - 9.05i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.37 - 8.86i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.15431435409917543233442307640, −10.57747083623131318420654419122, −9.365252228408945224819391082468, −8.519439136051442153356420212254, −7.66382922563491172039010990663, −6.88172324230669269684043500894, −6.38407227183192911491323093775, −5.09465114114146285471735882919, −4.40323077371258780738229629940, −2.79545316882566462129756438591, 0.16478861070945136109773034705, 1.13246476069625076167157097528, 2.78169598002326203819980918184, 3.88232394374456640984462384516, 4.81940249519274486331007197977, 6.03399183499098438223058497776, 7.39874215469106568805015985586, 8.312080728761865970352789397270, 9.177588290660964030980615075397, 9.817292520380780687657916615709

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