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Properties

Label 2-2717-2717.2430-c0-0-4
Degree $2$
Conductor $2717$
Sign $-0.996 - 0.0855i$
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  + (−0.194 + 1.10i)2-s + (1.47 + 1.23i)3-s + (−0.235 − 0.0857i)4-s + (−1.64 + 1.38i)6-s + (0.0348 + 0.0604i)7-s + (−0.418 + 0.725i)8-s + (0.468 + 2.65i)9-s + (0.5 − 0.866i)11-s + (−0.241 − 0.417i)12-s + (−0.766 + 0.642i)13-s + (−0.0733 + 0.0266i)14-s + (−0.909 − 0.763i)16-s − 3.01·18-s + (0.241 − 0.970i)19-s + (−0.0233 + 0.132i)21-s + (0.856 + 0.718i)22-s + ⋯
L(s)  = 1  + (−0.194 + 1.10i)2-s + (1.47 + 1.23i)3-s + (−0.235 − 0.0857i)4-s + (−1.64 + 1.38i)6-s + (0.0348 + 0.0604i)7-s + (−0.418 + 0.725i)8-s + (0.468 + 2.65i)9-s + (0.5 − 0.866i)11-s + (−0.241 − 0.417i)12-s + (−0.766 + 0.642i)13-s + (−0.0733 + 0.0266i)14-s + (−0.909 − 0.763i)16-s − 3.01·18-s + (0.241 − 0.970i)19-s + (−0.0233 + 0.132i)21-s + (0.856 + 0.718i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.996 - 0.0855i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (2430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ -0.996 - 0.0855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.876448894\)
\(L(\frac12)\) \(\approx\) \(1.876448894\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.241 + 0.970i)T \)
good2 \( 1 + (0.194 - 1.10i)T + (-0.939 - 0.342i)T^{2} \)
3 \( 1 + (-1.47 - 1.23i)T + (0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.0348 - 0.0604i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (1.86 + 0.677i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.35 - 1.13i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-1.35 - 0.492i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.23 - 1.04i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.615 + 1.06i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−9.130101761298355576040516781974, −8.562360921398147252494351113937, −8.078127166882226598108360620992, −7.30402332889262598778853581459, −6.47520011682501569567774256200, −5.50242616414117599136028149166, −4.60313581597504966791649550626, −3.98537709382718455452957929333, −2.82243939605949383917738065214, −2.28901006097455405022257786459, 1.11854225696499536659986067879, 1.99364605741043858926500135557, 2.52217607418903190036913092325, 3.52843761725134319587233583473, 4.06481959828869647620247591034, 5.73414522781198143516039890104, 6.64910980948465598262042762910, 7.40328235771816354511459882421, 7.82190988305360829021262566668, 8.760849045368090905057345351580

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