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Properties

Label 2-2808-2808.1507-c0-0-3
Degree $2$
Conductor $2808$
Sign $0.230 + 0.973i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
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  + (0.766 − 0.642i)2-s + (0.893 − 0.448i)3-s + (0.173 − 0.984i)4-s + (0.109 − 0.0397i)5-s + (0.396 − 0.918i)6-s + (0.207 + 1.17i)7-s + (−0.500 − 0.866i)8-s + (0.597 − 0.802i)9-s + (0.0581 − 0.100i)10-s + (−0.286 − 0.957i)12-s + (0.766 + 0.642i)13-s + (0.914 + 0.767i)14-s + (0.0798 − 0.0845i)15-s + (−0.939 − 0.342i)16-s + (0.286 − 0.496i)17-s + (−0.0581 − 0.998i)18-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.893 − 0.448i)3-s + (0.173 − 0.984i)4-s + (0.109 − 0.0397i)5-s + (0.396 − 0.918i)6-s + (0.207 + 1.17i)7-s + (−0.500 − 0.866i)8-s + (0.597 − 0.802i)9-s + (0.0581 − 0.100i)10-s + (−0.286 − 0.957i)12-s + (0.766 + 0.642i)13-s + (0.914 + 0.767i)14-s + (0.0798 − 0.0845i)15-s + (−0.939 − 0.342i)16-s + (0.286 − 0.496i)17-s + (−0.0581 − 0.998i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.653186459\)
\(L(\frac12)\) \(\approx\) \(2.653186459\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.893 + 0.448i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.109 + 0.0397i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.207 - 1.17i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.286 + 0.496i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.893 - 1.54i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (1.82 + 0.665i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.238 + 1.35i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.396 - 0.686i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−8.784124703206931105091936203753, −8.353837043764479076825217022528, −7.14412252839146661867369823515, −6.47186899339747576466679552949, −5.64722315591862443949553364758, −4.88013922135501858287453646265, −3.78238688248264972730017475368, −3.12406955494354369946142990860, −2.15328369361955835982701994410, −1.50567401801824365175759949107, 1.71128944440918932324335052613, 3.00298433010399945676720881433, 3.71502273206298616707194809730, 4.28304481068395838560748998743, 5.15490881924780048648751076212, 6.07258071800027587578930180276, 6.94491182033206946677162531466, 7.72846349674225155031361533003, 8.145124334667972385685403325166, 8.889206156659653251659959715504

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