Properties

Label 2-2151-1.1-c1-0-23
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0888·2-s − 1.99·4-s + 0.674·5-s + 0.687·7-s − 0.354·8-s + 0.0599·10-s + 0.536·11-s − 0.628·13-s + 0.0611·14-s + 3.95·16-s − 5.05·17-s − 0.724·19-s − 1.34·20-s + 0.0477·22-s + 7.36·23-s − 4.54·25-s − 0.0558·26-s − 1.37·28-s + 3.61·29-s + 5.74·31-s + 1.06·32-s − 0.449·34-s + 0.463·35-s − 0.110·37-s − 0.0644·38-s − 0.239·40-s − 8.05·41-s + ⋯
L(s)  = 1  + 0.0628·2-s − 0.996·4-s + 0.301·5-s + 0.260·7-s − 0.125·8-s + 0.0189·10-s + 0.161·11-s − 0.174·13-s + 0.0163·14-s + 0.988·16-s − 1.22·17-s − 0.166·19-s − 0.300·20-s + 0.0101·22-s + 1.53·23-s − 0.909·25-s − 0.0109·26-s − 0.258·28-s + 0.671·29-s + 1.03·31-s + 0.187·32-s − 0.0770·34-s + 0.0784·35-s − 0.0182·37-s − 0.0104·38-s − 0.0378·40-s − 1.25·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435458817\)
\(L(\frac12)\) \(\approx\) \(1.435458817\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 - 0.0888T + 2T^{2} \)
5 \( 1 - 0.674T + 5T^{2} \)
7 \( 1 - 0.687T + 7T^{2} \)
11 \( 1 - 0.536T + 11T^{2} \)
13 \( 1 + 0.628T + 13T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 0.724T + 19T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
37 \( 1 + 0.110T + 37T^{2} \)
41 \( 1 + 8.05T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 - 9.81T + 59T^{2} \)
61 \( 1 - 6.30T + 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 - 8.10T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−8.895247987715656915826930940702, −8.605832345099405287734300675809, −7.59396934191499772631941251495, −6.69530165544076608208974425068, −5.84850950681180972265743930560, −4.89840613528370836121914020695, −4.40919668339774802539130757061, −3.34619109400038917775928488387, −2.19147804479805430018169871320, −0.792898479980978233185509339486, 0.792898479980978233185509339486, 2.19147804479805430018169871320, 3.34619109400038917775928488387, 4.40919668339774802539130757061, 4.89840613528370836121914020695, 5.84850950681180972265743930560, 6.69530165544076608208974425068, 7.59396934191499772631941251495, 8.605832345099405287734300675809, 8.895247987715656915826930940702

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