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Properties

Label 2-1380-1380.1259-c0-0-1
Degree $2$
Conductor $1380$
Sign $-0.403 - 0.915i$
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  + (−0.989 − 0.142i)2-s + (−0.281 + 0.959i)3-s + (0.959 + 0.281i)4-s + (0.540 + 0.841i)5-s + (0.415 − 0.909i)6-s + (−0.909 − 0.415i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)10-s + (−0.540 + 0.841i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (0.989 + 0.857i)17-s + (0.755 + 0.654i)18-s + (0.186 + 0.215i)19-s + (0.281 + 0.959i)20-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.281 + 0.959i)3-s + (0.959 + 0.281i)4-s + (0.540 + 0.841i)5-s + (0.415 − 0.909i)6-s + (−0.909 − 0.415i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)10-s + (−0.540 + 0.841i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (0.989 + 0.857i)17-s + (0.755 + 0.654i)18-s + (0.186 + 0.215i)19-s + (0.281 + 0.959i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.403 - 0.915i$
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: \((2,\ 1380,\ (\ :0),\ -0.403 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6436552798\)
\(L(\frac12)\) \(\approx\) \(0.6436552798\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 + 0.142i)T \)
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

−10.05981317531458937756450252301, −9.520370728922123205415009714886, −8.544177666035973828132921082837, −7.81541423600741605023747117234, −6.70175053243265172177749654427, −6.07364890467420421466885286431, −5.20931458847559574711442426044, −3.66781413705781745566072192142, −3.06757751943100905601618131468, −1.70601316779693346808504191428, 0.77675366221029648630479549200, 1.87873824757950734773047908811, 2.87153747814947029730983796928, 4.71670132346117829273010500338, 5.81889938307694167384779505718, 6.16715796239147213799966737168, 7.34682220414954683930926035551, 7.891393113305773517162213415009, 8.622225044946990304837027504070, 9.507123126682662988404672937817

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