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Properties

Label 2-1380-1380.419-c0-0-0
Degree $2$
Conductor $1380$
Sign $0.525 - 0.850i$
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  + (0.281 − 0.959i)2-s + (0.540 + 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (−0.755 + 0.345i)7-s + (−0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.909 + 0.415i)10-s − i·12-s + (0.118 + 0.822i)14-s + (−0.909 + 0.415i)15-s + (0.415 + 0.909i)16-s + (0.755 + 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)2-s + (0.540 + 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (−0.755 + 0.345i)7-s + (−0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.909 + 0.415i)10-s − i·12-s + (0.118 + 0.822i)14-s + (−0.909 + 0.415i)15-s + (0.415 + 0.909i)16-s + (0.755 + 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051719325\)
\(L(\frac12)\) \(\approx\) \(1.051719325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 + 0.959i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
good7 \( 1 + (0.755 - 0.345i)T + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 - 0.284iT - T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.368 + 1.25i)T + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.215 + 1.49i)T + (-0.959 + 0.281i)T^{2} \)
89 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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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

−9.996785288089686955047692362075, −9.373788996781760868197402047855, −8.644330844921953702749266790954, −7.63838207815974797310603781142, −6.46538636174240286194562462743, −5.55546699084097840384811859097, −4.58425652552289069983547551624, −3.47314892170245885874786405908, −3.13232961717978621977461544510, −2.08386410629134294654506980603, 0.75413028287659331082273010859, 2.59311854222325359996865254613, 3.79511666777068439160657513070, 4.54356062171326277866463196240, 5.75821237339789468181660327575, 6.42912051300754939292751923197, 7.18364589049538352820981577335, 8.150346878357362391921565854064, 8.428123920941942104293145169275, 9.440541901531062929692693387372

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