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Properties

Label 2-1380-1380.359-c0-0-3
Degree $2$
Conductor $1380$
Sign $0.159 - 0.987i$
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  + (0.909 + 0.415i)2-s + (−0.755 + 0.654i)3-s + (0.654 + 0.755i)4-s + (0.989 − 0.142i)5-s + (−0.959 + 0.281i)6-s + (0.281 + 0.959i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)10-s + (−0.989 − 0.142i)12-s + (−0.654 + 0.755i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 1.41i)17-s + (0.540 − 0.841i)18-s + (0.698 − 0.449i)19-s + (0.755 + 0.654i)20-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (−0.755 + 0.654i)3-s + (0.654 + 0.755i)4-s + (0.989 − 0.142i)5-s + (−0.959 + 0.281i)6-s + (0.281 + 0.959i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)10-s + (−0.989 − 0.142i)12-s + (−0.654 + 0.755i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 1.41i)17-s + (0.540 − 0.841i)18-s + (0.698 − 0.449i)19-s + (0.755 + 0.654i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.702111750\)
\(L(\frac12)\) \(\approx\) \(1.702111750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 - 0.415i)T \)
3 \( 1 + (0.755 - 0.654i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
23 \( 1 + (-0.281 + 0.959i)T \)
good7 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (0.909 - 1.41i)T + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.415 - 0.909i)T^{2} \)
31 \( 1 + (1.37 + 1.19i)T + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.959 - 0.281i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.142 - 0.989i)T^{2} \)
47 \( 1 - 1.91iT - T^{2} \)
53 \( 1 + (1.74 + 0.797i)T + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.425 - 0.368i)T + (0.142 + 0.989i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (-0.215 + 1.49i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.142 + 0.989i)T^{2} \)
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

−10.03350918506062950373787078471, −9.206860264918749177654162610385, −8.387006220362710576812231513238, −7.15282933722899709790493402674, −6.27942341867022313063179919794, −5.88553099208171364699703323145, −4.93507184461083917332952191345, −4.29678381926428295317233497240, −3.18837304230988450899383410123, −1.89033792249303889225798080602, 1.36635764966846977664107319026, 2.30189539363183973687411219214, 3.38279868236033452294047202971, 4.94013167194391144526728518004, 5.26872052822808582137330530897, 6.18588430074037569554844520408, 6.90987354410814484668949855303, 7.51777935441074721809665694092, 9.074689183829569506791279269075, 9.787090766530453380297860695431

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