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Properties

Label 2-1380-1380.839-c0-0-1
Degree $2$
Conductor $1380$
Sign $-0.0654 - 0.997i$
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 + (−0.989 − 0.142i)3-s + (0.142 − 0.989i)4-s + (0.281 + 0.959i)5-s + (0.841 − 0.540i)6-s + (0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)10-s + (−0.281 + 0.959i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (0.755 − 0.345i)17-s + (−0.909 + 0.415i)18-s + (−0.544 + 1.19i)19-s + (0.989 − 0.142i)20-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (−0.989 − 0.142i)3-s + (0.142 − 0.989i)4-s + (0.281 + 0.959i)5-s + (0.841 − 0.540i)6-s + (0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)10-s + (−0.281 + 0.959i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (0.755 − 0.345i)17-s + (−0.909 + 0.415i)18-s + (−0.544 + 1.19i)19-s + (0.989 − 0.142i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0654 - 0.997i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -0.0654 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5579634341\)
\(L(\frac12)\) \(\approx\) \(0.5579634341\)
\(L(1)\) 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 \( 1 + (0.989 + 0.142i)T \)
5 \( 1 + (-0.281 - 0.959i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
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 \( 1 + (-0.755 + 0.345i)T + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
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 \( 1 - 1.68iT - T^{2} \)
53 \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2} \)
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 \( 1 + (-1.89 - 0.557i)T + (0.841 + 0.540i)T^{2} \)
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

−10.04714295112230554821681564047, −9.382652201562331356760512366792, −8.078885286097677235739821161752, −7.54753555186015960751204008527, −6.52876409973761510140501475268, −6.21776658104563226621153850356, −5.33358750446299411154841082812, −4.29748153294518533191978231049, −2.67279301325216371979197014575, −1.31052850057348970742833483500, 0.78013692954593520188534338256, 1.92396805846462193651753545083, 3.45182561978225815309448757253, 4.54285062351812153769900753360, 5.23143987499839356894013279964, 6.34936306711045908531042890061, 7.17286967312797324524649902351, 8.193484206199597210154094152472, 8.900098994450919775592676475854, 9.767417633926379323777858603131

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