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Properties

Label 2-1380-1380.1019-c0-0-5
Degree $2$
Conductor $1380$
Sign $-0.167 + 0.985i$
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 + (−0.540 − 0.841i)3-s + (0.959 + 0.281i)4-s + (0.909 − 0.415i)5-s + (0.415 + 0.909i)6-s + (−0.909 − 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (−0.281 − 0.959i)12-s + (−0.841 − 0.540i)15-s + (0.841 + 0.540i)16-s + (−0.281 − 0.0405i)17-s + (0.540 − 0.841i)18-s + (−0.273 − 1.89i)19-s + (0.989 − 0.142i)20-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.540 − 0.841i)3-s + (0.959 + 0.281i)4-s + (0.909 − 0.415i)5-s + (0.415 + 0.909i)6-s + (−0.909 − 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (−0.281 − 0.959i)12-s + (−0.841 − 0.540i)15-s + (0.841 + 0.540i)16-s + (−0.281 − 0.0405i)17-s + (0.540 − 0.841i)18-s + (−0.273 − 1.89i)19-s + (0.989 − 0.142i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.167 + 0.985i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -0.167 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6490885706\)
\(L(\frac12)\) \(\approx\) \(0.6490885706\)
\(L(1)\) 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 \( 1 + (0.540 + 0.841i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
23 \( 1 + (-0.755 + 0.654i)T \)
good7 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.281 + 0.0405i)T + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - 1.30iT - T^{2} \)
53 \( 1 + (0.368 + 1.25i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.817 + 1.27i)T + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.449 - 0.983i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.654 + 0.755i)T^{2} \)
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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

−9.391887809397564942643296414243, −8.818762466343192270300341109479, −8.067468863729801989359653043817, −6.88448903244348851645852945396, −6.68154489916735442246129993477, −5.60431657858491071430970387892, −4.74206593006363608855338267104, −2.83030027337197572264875632950, −2.04033792919189885890668540913, −0.832052279901444240169861029817, 1.52685587314587418990601557877, 2.81667026418394936151930437020, 3.93057625382645842199638544767, 5.40551909832488514069668983977, 5.87352072365062387268150291821, 6.67743264505803140412515251215, 7.58797164083192641153282888584, 8.703960620952681225346660109023, 9.293488930656173356014103783680, 10.11922651951114379486470018554

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