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Properties

Label 2-639-71.5-c1-0-14
Degree $2$
Conductor $639$
Sign $-0.0922 - 0.995i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
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  + (0.472 + 1.45i)2-s + (−0.274 + 0.199i)4-s + (2.89 + 2.10i)5-s + (−0.0936 − 0.288i)7-s + (2.05 + 1.49i)8-s + (−1.69 + 5.20i)10-s + (0.453 + 1.39i)11-s + (0.961 − 2.95i)13-s + (0.375 − 0.272i)14-s + (−1.40 + 4.33i)16-s + (0.448 − 1.38i)17-s + (−2.31 − 7.11i)19-s − 1.21·20-s + (−1.81 + 1.31i)22-s − 3.26·23-s + ⋯
L(s)  = 1  + (0.334 + 1.02i)2-s + (−0.137 + 0.0995i)4-s + (1.29 + 0.940i)5-s + (−0.0354 − 0.108i)7-s + (0.726 + 0.527i)8-s + (−0.534 + 1.64i)10-s + (0.136 + 0.420i)11-s + (0.266 − 0.820i)13-s + (0.100 − 0.0728i)14-s + (−0.352 + 1.08i)16-s + (0.108 − 0.334i)17-s + (−0.530 − 1.63i)19-s − 0.271·20-s + (−0.387 + 0.281i)22-s − 0.681·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0922 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0922 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.0922 - 0.995i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.0922 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62847 + 1.78625i\)
\(L(\frac12)\) \(\approx\) \(1.62847 + 1.78625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (5.01 + 6.76i)T \)
good2 \( 1 + (-0.472 - 1.45i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-2.89 - 2.10i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.0936 + 0.288i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.453 - 1.39i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.961 + 2.95i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.448 + 1.38i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.31 + 7.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 + (2.30 + 1.67i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.84 - 8.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 - 3.94T + 37T^{2} \)
41 \( 1 + 6.62T + 41T^{2} \)
43 \( 1 + (9.64 + 7.00i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (3.66 - 11.2i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.56 + 1.86i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.75 + 3.45i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.949 - 2.92i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.23 + 2.35i)T + (20.7 - 63.7i)T^{2} \)
73 \( 1 + (0.674 + 2.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.73 + 1.99i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.83 - 4.23i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-9.35 - 6.79i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + 7.89T + 97T^{2} \)
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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

−10.58870038326398165775526508597, −10.08882019764883104990373255230, −9.019243124673330521831019971621, −7.905051628517477107614804315436, −6.85632463086883179746923002400, −6.50491455324068340519564287733, −5.55061984787619593375233928890, −4.73732734335247370811980120942, −3.03297755580794853224226936048, −1.89867521442487758331813603355, 1.45286778291352850350144950730, 2.13979137451109620319671407531, 3.63335581518886960934577942727, 4.55869181203176584480348363287, 5.76085534411021997324953557478, 6.43883685276148386251772258897, 7.939452136582311247431537893100, 8.823737440674921247105231681963, 9.848472779610052016265585328734, 10.17786022102416887481746882864

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