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Properties

Label 2-2717-2717.1715-c0-0-0
Degree $2$
Conductor $2717$
Sign $0.305 - 0.952i$
Analytic cond. $1.35595$
Root an. cond. $1.16445$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 + 0.524i)2-s + (0.266 − 1.50i)3-s + (1.03 − 0.866i)4-s + (0.407 + 2.31i)6-s + (−0.766 − 1.32i)7-s + (−0.266 + 0.460i)8-s + (−1.26 − 0.460i)9-s + (−0.5 + 0.866i)11-s + (−1.03 − 1.78i)12-s + (0.173 + 0.984i)13-s + (1.79 + 1.50i)14-s + (−0.0923 + 0.524i)16-s + 2.06·18-s + (0.173 + 0.984i)19-s + (−2.20 + 0.802i)21-s + (0.266 − 1.50i)22-s + ⋯
L(s)  = 1  + (−1.43 + 0.524i)2-s + (0.266 − 1.50i)3-s + (1.03 − 0.866i)4-s + (0.407 + 2.31i)6-s + (−0.766 − 1.32i)7-s + (−0.266 + 0.460i)8-s + (−1.26 − 0.460i)9-s + (−0.5 + 0.866i)11-s + (−1.03 − 1.78i)12-s + (0.173 + 0.984i)13-s + (1.79 + 1.50i)14-s + (−0.0923 + 0.524i)16-s + 2.06·18-s + (0.173 + 0.984i)19-s + (−2.20 + 0.802i)21-s + (0.266 − 1.50i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (1715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2543236860\)
\(L(\frac12)\) \(\approx\) \(0.2543236860\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
3 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−9.131725915757792750790879942497, −7.970679163088282038218895577494, −7.72346625362794789463849731109, −7.18455302767378791611923194200, −6.54809324628231096117493699971, −5.98355542513511998825280870732, −4.36930867356654194524366287060, −3.32808702173039420378500045862, −1.81948529956833998927977698281, −1.34713711105153704276520840481, 0.25825377350306310823316308621, 2.43740769611813995763732286311, 2.82782283699785313243963464751, 3.77297771508946813490430061656, 5.02430100370841554165741927350, 5.66223300741925870222705627285, 6.60857653794672932971197974446, 7.975286954269110415174324947842, 8.606755071802292291229439524573, 8.808492701906820012738760100227

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