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Properties

Label 2-1380-1380.779-c0-0-3
Degree $2$
Conductor $1380$
Sign $0.999 - 0.0137i$
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 + (0.909 + 0.415i)3-s + (−0.415 + 0.909i)4-s + (0.755 + 0.654i)5-s + (−0.142 − 0.989i)6-s + (0.989 − 0.142i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)10-s + (−0.755 + 0.654i)12-s + (0.415 + 0.909i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 1.84i)17-s + (0.281 − 0.959i)18-s + (−1.61 + 0.474i)19-s + (−0.909 + 0.415i)20-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (0.909 + 0.415i)3-s + (−0.415 + 0.909i)4-s + (0.755 + 0.654i)5-s + (−0.142 − 0.989i)6-s + (0.989 − 0.142i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)10-s + (−0.755 + 0.654i)12-s + (0.415 + 0.909i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 1.84i)17-s + (0.281 − 0.959i)18-s + (−1.61 + 0.474i)19-s + (−0.909 + 0.415i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0137i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.999 - 0.0137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235891749\)
\(L(\frac12)\) \(\approx\) \(1.235891749\)
\(L(1)\) 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 \( 1 + (-0.909 - 0.415i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
good7 \( 1 + (-0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.540 + 1.84i)T + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.983 - 0.449i)T + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 + 0.284iT - T^{2} \)
53 \( 1 + (-0.153 - 0.239i)T + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (1.80 - 0.822i)T + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (1.19 + 1.37i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.142 - 0.989i)T^{2} \)
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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

−9.700861513819088756167862340723, −9.181979351585543849449919037706, −8.500074641361582407553497672118, −7.44898899392761418597560296099, −6.91012143065843412597402602207, −5.40630781545527489534915825945, −4.42288187896433487145460670899, −3.31616004102099955462028965114, −2.67034235501626597056442182920, −1.71877173131364251103765454546, 1.34131330051587332243974591869, 2.24780729396689834654220839488, 3.88887070901806041916838386997, 4.82429965677008361381380925331, 6.00659364934111803166950206045, 6.46178902389421309554979240640, 7.52235501141626450669696998938, 8.290434763199087572419937786203, 8.851062339963067194084271440084, 9.386049440721987109165884987244

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