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Properties

Label 2-1380-1380.359-c0-0-7
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: $\chi_{1380} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.911601623\)
\(L(\frac12)\) \(\approx\) \(1.911601623\)
\(L(1)\) 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

−9.692442845928838812335854691631, −8.694327968753511490336799190423, −8.298914073264932852764279003910, −6.72354984070560108787948112340, −6.28345292488409834428619372028, −5.29598879956987603002808789614, −4.16880888113910354756801206126, −3.24115141382537317918521450525, −2.12591924375118151141751664417, −1.52900475071445456830066682284, 2.40143048139237154315328407795, 2.97691912364608196714954239650, 4.49861186488070511873580481759, 4.72256981848983171044082677677, 5.98803753806924201330981606515, 6.63110097451866948695452360833, 7.61833860735576161732749769217, 8.432975981674939287835580248493, 9.256407982210935430855627061947, 9.625175606863439935620033095863

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