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Properties

Label 2-1380-1380.1259-c0-0-3
Degree $2$
Conductor $1380$
Sign $0.664 - 0.746i$
Analytic cond. $0.688709$
Root an. cond. $0.829885$
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  + (0.755 + 0.654i)2-s + (0.281 − 0.959i)3-s + (0.142 + 0.989i)4-s + (0.540 + 0.841i)5-s + (0.841 − 0.540i)6-s + (−0.540 + 0.841i)8-s + (−0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (0.989 + 0.142i)12-s + (0.959 − 0.281i)15-s + (−0.959 + 0.281i)16-s + (0.989 + 0.857i)17-s + (−0.281 − 0.959i)18-s + (−0.186 − 0.215i)19-s + (−0.755 + 0.654i)20-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (0.281 − 0.959i)3-s + (0.142 + 0.989i)4-s + (0.540 + 0.841i)5-s + (0.841 − 0.540i)6-s + (−0.540 + 0.841i)8-s + (−0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (0.989 + 0.142i)12-s + (0.959 − 0.281i)15-s + (−0.959 + 0.281i)16-s + (0.989 + 0.857i)17-s + (−0.281 − 0.959i)18-s + (−0.186 − 0.215i)19-s + (−0.755 + 0.654i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.664 - 0.746i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.854135600\)
\(L(\frac12)\) \(\approx\) \(1.854135600\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + (-0.281 + 0.959i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (-0.909 + 0.415i)T \)
good7 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (-0.989 - 0.857i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.186 + 0.215i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - 0.830iT - T^{2} \)
53 \( 1 + (0.822 + 0.118i)T + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.512 + 1.74i)T + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.474 - 0.304i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.415 - 0.909i)T^{2} \)
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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

−9.724930307138941556522400934175, −8.858454505354407939329385204578, −7.84625879130299242177874400627, −7.45895828687291083368011604683, −6.37294122700829168876039389941, −6.13827996140074811804367899839, −5.09160377703950373312500550277, −3.69412815103295903269205508925, −2.91286459463640534245821495800, −1.91381860779690225384430606729, 1.42712789326257123769151977747, 2.77779698003615073013619819330, 3.58077665961311703606053592717, 4.68680651989709913482067187452, 5.19502826000080453663636886850, 5.87640201136271278664408806533, 7.12008881565896073922041873746, 8.410963840126898122524599437401, 9.169229054974843435038381301537, 9.737537309601280755562625424306

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