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Properties

Label 2-3513-3513.2891-c0-0-0
Degree $2$
Conductor $3513$
Sign $-0.785 - 0.619i$
Analytic cond. $1.75321$
Root an. cond. $1.32409$
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  + (−0.919 + 0.391i)3-s + (0.692 + 0.721i)4-s + (−1.47 − 1.10i)7-s + (0.692 − 0.721i)9-s + (−0.919 − 0.391i)12-s + (0.213 + 1.75i)13-s + (−0.0402 + 0.999i)16-s + (0.103 − 0.217i)19-s + (1.78 + 0.440i)21-s + (−0.5 − 0.866i)25-s + (−0.354 + 0.935i)27-s + (−0.221 − 1.82i)28-s + (0.253 + 1.23i)31-s + 36-s + (−1.35 + 0.854i)37-s + ⋯
L(s)  = 1  + (−0.919 + 0.391i)3-s + (0.692 + 0.721i)4-s + (−1.47 − 1.10i)7-s + (0.692 − 0.721i)9-s + (−0.919 − 0.391i)12-s + (0.213 + 1.75i)13-s + (−0.0402 + 0.999i)16-s + (0.103 − 0.217i)19-s + (1.78 + 0.440i)21-s + (−0.5 − 0.866i)25-s + (−0.354 + 0.935i)27-s + (−0.221 − 1.82i)28-s + (0.253 + 1.23i)31-s + 36-s + (−1.35 + 0.854i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3513\)    =    \(3 \cdot 1171\)
Sign: $-0.785 - 0.619i$
Analytic conductor: \(1.75321\)
Root analytic conductor: \(1.32409\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3513} (2891, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3513,\ (\ :0),\ -0.785 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5552676685\)
\(L(\frac12)\) \(\approx\) \(0.5552676685\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.919 - 0.391i)T \)
1171 \( 1 + (-0.120 + 0.992i)T \)
good2 \( 1 + (-0.692 - 0.721i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (1.47 + 1.10i)T + (0.278 + 0.960i)T^{2} \)
11 \( 1 + (-0.120 + 0.992i)T^{2} \)
13 \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \)
17 \( 1 + (0.632 + 0.774i)T^{2} \)
19 \( 1 + (-0.103 + 0.217i)T + (-0.632 - 0.774i)T^{2} \)
23 \( 1 + (0.748 + 0.663i)T^{2} \)
29 \( 1 + (0.200 + 0.979i)T^{2} \)
31 \( 1 + (-0.253 - 1.23i)T + (-0.919 + 0.391i)T^{2} \)
37 \( 1 + (1.35 - 0.854i)T + (0.428 - 0.903i)T^{2} \)
41 \( 1 + (-0.568 - 0.822i)T^{2} \)
43 \( 1 + (1.35 + 0.854i)T + (0.428 + 0.903i)T^{2} \)
47 \( 1 + (-0.885 + 0.464i)T^{2} \)
53 \( 1 + (-0.120 + 0.992i)T^{2} \)
59 \( 1 + (-0.428 + 0.903i)T^{2} \)
61 \( 1 + (-0.0680 - 1.68i)T + (-0.996 + 0.0804i)T^{2} \)
67 \( 1 - 1.89T + T^{2} \)
71 \( 1 + (0.632 - 0.774i)T^{2} \)
73 \( 1 + (-0.759 - 1.59i)T + (-0.632 + 0.774i)T^{2} \)
79 \( 1 + (1.19 - 0.899i)T + (0.278 - 0.960i)T^{2} \)
83 \( 1 + (0.845 - 0.534i)T^{2} \)
89 \( 1 + (-0.692 + 0.721i)T^{2} \)
97 \( 1 + (1.84 + 0.453i)T + (0.885 + 0.464i)T^{2} \)
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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

−9.145085499075824955508950868444, −8.320519547990462647331618767943, −7.04899684187518705993473666842, −6.77203021265389659031150950629, −6.49374799721266960277508020870, −5.31138371251782837505931644870, −4.07817144955059945618485036241, −3.89499051816023537188781385898, −2.85739150957188879299298975020, −1.45524929398936892313225482632, 0.35366160294823422248292040763, 1.77990022189920757002896111777, 2.75820734452590167770103325756, 3.56877114803798169198704568649, 5.24549076609862922854411165186, 5.50761171482354840885308390077, 6.19995656914071276298182479485, 6.68206269768651563899922949485, 7.56299638808443746432957663599, 8.334366765197995453663353095030

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