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Properties

Label 2-1380-1380.419-c0-0-1
Degree $2$
Conductor $1380$
Sign $0.431 + 0.902i$
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  + (−0.540 − 0.841i)2-s + (−0.755 − 0.654i)3-s + (−0.415 + 0.909i)4-s + (−0.989 − 0.142i)5-s + (−0.142 + 0.989i)6-s + (0.989 − 0.142i)8-s + (0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (0.909 − 0.415i)12-s + (0.654 + 0.755i)15-s + (−0.654 − 0.755i)16-s + (0.909 + 1.41i)17-s + (0.755 − 0.654i)18-s + (−0.698 − 0.449i)19-s + (0.540 − 0.841i)20-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (−0.755 − 0.654i)3-s + (−0.415 + 0.909i)4-s + (−0.989 − 0.142i)5-s + (−0.142 + 0.989i)6-s + (0.989 − 0.142i)8-s + (0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (0.909 − 0.415i)12-s + (0.654 + 0.755i)15-s + (−0.654 − 0.755i)16-s + (0.909 + 1.41i)17-s + (0.755 − 0.654i)18-s + (−0.698 − 0.449i)19-s + (0.540 − 0.841i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.431 + 0.902i$
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: \((2,\ 1380,\ (\ :0),\ 0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4930439863\)
\(L(\frac12)\) \(\approx\) \(0.4930439863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.540 + 0.841i)T \)
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

−9.889059654659187811978240427103, −8.626009770710630351303700764982, −8.104644251713836229137506977164, −7.42527548052728364795846406654, −6.54498440506649157054120177112, −5.37495787820613758055873895802, −4.34688408607196105984696920705, −3.51712363665045832546584142793, −2.16557732985315296829710647571, −0.903674413345020090324058136489, 0.822320117411346227358559751189, 3.08158432015967893510661214340, 4.36502135350326322863420428874, 4.86520620940660865350705289116, 5.88411083020321655461796486536, 6.73711934419762213777211170076, 7.39116370291187688911704694574, 8.346075207346664534958495161286, 9.004610979536889636621234905871, 10.00470395085615752638739141172

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