Properties

Label 2-2760-2760.2219-c0-0-3
Degree $2$
Conductor $2760$
Sign $-0.354 + 0.934i$
Analytic cond. $1.37741$
Root an. cond. $1.17363$
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  + (−0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)10-s + (0.959 + 0.281i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (1.65 − 0.755i)17-s + (0.415 + 0.909i)18-s + (1.37 + 0.627i)19-s + (0.142 + 0.989i)20-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)10-s + (0.959 + 0.281i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (1.65 − 0.755i)17-s + (0.415 + 0.909i)18-s + (1.37 + 0.627i)19-s + (0.142 + 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.354 + 0.934i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (2219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ -0.354 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142013216\)
\(L(\frac12)\) \(\approx\) \(1.142013216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good7 \( 1 + (0.142 - 0.989i)T^{2} \)
11 \( 1 + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (-1.65 + 0.755i)T + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-1.37 - 0.627i)T + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (-0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - 1.08iT - T^{2} \)
53 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.557 + 1.89i)T + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.540i)T^{2} \)
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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

−8.994580202080264977143372973859, −7.88435866167206753347694226961, −7.57464777023880488142293148231, −6.69652068727261095983490329389, −5.64454829770204505739231365794, −5.06307582179603976824180796502, −3.41326219846391613915295040935, −2.91588692908030763920200763355, −1.70187520718880059204330783762, −1.10245396466966919438288208418, 1.35249137706804468664053459871, 2.68221531761419465860553469974, 3.64220934666856922841544830401, 4.93026487947649144467539933646, 5.46899618718725283061174707311, 6.01372257294055893700213794027, 7.05974137464429443038497412505, 7.73309494624639151902362477853, 8.703737804169050293516885754139, 9.234301243675967860623976019148

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