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Properties

Label 2-2717-2717.2144-c0-0-2
Degree $2$
Conductor $2717$
Sign $0.944 + 0.327i$
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  + (−0.266 − 0.223i)2-s + (−0.326 + 0.118i)3-s + (−0.152 − 0.866i)4-s + (0.113 + 0.0412i)6-s + (0.173 + 0.300i)7-s + (−0.326 + 0.565i)8-s + (−0.673 + 0.565i)9-s + (0.5 − 0.866i)11-s + (0.152 + 0.264i)12-s + (0.939 + 0.342i)13-s + (0.0209 − 0.118i)14-s + (−0.613 + 0.223i)16-s + 0.305·18-s + (0.939 + 0.342i)19-s + (−0.0923 − 0.0775i)21-s + (−0.326 + 0.118i)22-s + ⋯
L(s)  = 1  + (−0.266 − 0.223i)2-s + (−0.326 + 0.118i)3-s + (−0.152 − 0.866i)4-s + (0.113 + 0.0412i)6-s + (0.173 + 0.300i)7-s + (−0.326 + 0.565i)8-s + (−0.673 + 0.565i)9-s + (0.5 − 0.866i)11-s + (0.152 + 0.264i)12-s + (0.939 + 0.342i)13-s + (0.0209 − 0.118i)14-s + (−0.613 + 0.223i)16-s + 0.305·18-s + (0.939 + 0.342i)19-s + (−0.0923 − 0.0775i)21-s + (−0.326 + 0.118i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (2144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9160103595\)
\(L(\frac12)\) \(\approx\) \(0.9160103595\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good2 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−9.098675947580248459391860344530, −8.391874681161248642863279558529, −7.61474651907133111630695711143, −6.41516828230155677217861529453, −5.62848371495096249571900634851, −5.51925837929477895904220793314, −4.26031499739809508911773476370, −3.27400636553587453739885773878, −2.07725503484595705683853139867, −1.03311536749421399291740911425, 0.906415992510992532547876469093, 2.51460193970700431679135380473, 3.51732829182381134837280990178, 4.19183661958340061574777444094, 5.19010619733507979996785902916, 6.24360859858196397597112329272, 6.76883894552392122855905220087, 7.61592146411482419503824680596, 8.229198777248654013541190739065, 9.060382042209138818170713811942

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