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Properties

Label 2-2717-2717.2144-c0-0-5
Degree $2$
Conductor $2717$
Sign $-0.0197 + 0.999i$
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  + (−1.51 − 1.27i)2-s + (1.65 − 0.603i)3-s + (0.507 + 2.87i)4-s + (−3.28 − 1.19i)6-s + (0.438 + 0.759i)7-s + (1.90 − 3.29i)8-s + (1.62 − 1.36i)9-s + (0.5 − 0.866i)11-s + (2.58 + 4.46i)12-s + (0.939 + 0.342i)13-s + (0.301 − 1.71i)14-s + (−4.34 + 1.57i)16-s − 4.19·18-s + (−0.0348 + 0.999i)19-s + (1.18 + 0.995i)21-s + (−1.86 + 0.677i)22-s + ⋯
L(s)  = 1  + (−1.51 − 1.27i)2-s + (1.65 − 0.603i)3-s + (0.507 + 2.87i)4-s + (−3.28 − 1.19i)6-s + (0.438 + 0.759i)7-s + (1.90 − 3.29i)8-s + (1.62 − 1.36i)9-s + (0.5 − 0.866i)11-s + (2.58 + 4.46i)12-s + (0.939 + 0.342i)13-s + (0.301 − 1.71i)14-s + (−4.34 + 1.57i)16-s − 4.19·18-s + (−0.0348 + 0.999i)19-s + (1.18 + 0.995i)21-s + (−1.86 + 0.677i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0197 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0197 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.0197 + 0.999i$
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: \((2,\ 2717,\ (\ :0),\ -0.0197 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184792443\)
\(L(\frac12)\) \(\approx\) \(1.184792443\)
\(L(1)\) 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 \( 1 + (0.0348 - 0.999i)T \)
good2 \( 1 + (1.51 + 1.27i)T + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (-1.65 + 0.603i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.438 - 0.759i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.0840 + 0.476i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.87 - 0.682i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.294 - 1.67i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.719 + 1.24i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.817197670036588066725521010787, −8.239394189928791790820429542131, −8.041859752662737849229604268695, −7.02736461952691270527820874130, −6.17492165756744789834158386594, −4.12700524547830119060676895434, −3.51779669081953712567033327334, −2.78567131455508602426749107009, −1.89594222994817302586147903687, −1.33362841317408429065041531930, 1.38495178815640780141329103011, 2.18807862261913893273433617060, 3.67527317192001870076697951758, 4.58751207523737770724268750034, 5.43978832204505388858705068582, 6.73192304791261561929046606751, 7.19394201983453175760688458704, 7.947477478634272671887695633465, 8.389455183431114349017231930926, 9.061852464091255967128265018409

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