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Properties

Label 2-1380-1380.659-c0-0-3
Degree $2$
Conductor $1380$
Sign $0.764 - 0.644i$
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  + (−0.989 − 0.142i)2-s + (0.540 + 0.841i)3-s + (0.959 + 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.415 − 0.909i)6-s + (0.909 − 1.41i)7-s + (−0.909 − 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.540 − 0.841i)10-s + (0.281 + 0.959i)12-s + (−1.10 + 1.27i)14-s + (−0.281 + 0.959i)15-s + (0.841 + 0.540i)16-s + (0.540 − 0.841i)18-s + (0.415 + 0.909i)20-s + 1.68·21-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (0.540 + 0.841i)3-s + (0.959 + 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.415 − 0.909i)6-s + (0.909 − 1.41i)7-s + (−0.909 − 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.540 − 0.841i)10-s + (0.281 + 0.959i)12-s + (−1.10 + 1.27i)14-s + (−0.281 + 0.959i)15-s + (0.841 + 0.540i)16-s + (0.540 − 0.841i)18-s + (0.415 + 0.909i)20-s + 1.68·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033719037\)
\(L(\frac12)\) \(\approx\) \(1.033719037\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
good7 \( 1 + (-0.909 + 1.41i)T + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.425 + 1.45i)T + (-0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (-0.142 - 0.989i)T^{2} \)
41 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
43 \( 1 + (1.74 - 0.797i)T + (0.654 - 0.755i)T^{2} \)
47 \( 1 - 1.30iT - T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.512 + 0.234i)T + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.822 + 0.118i)T + (0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
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

−9.854191324290929517334209635563, −9.303672398694141613822645030002, −8.253431656396789401639078844963, −7.62396865430253821384645586489, −6.95061249274749129415215756075, −5.84945818756003122694323241599, −4.64048977152823234644497032288, −3.64013606823463289239241707401, −2.67842261873593706692179414327, −1.54188935139499269351434610562, 1.39013300323583246754015386871, 2.07625891181847756104284303111, 3.04108490223247016228024918266, 5.03357318092286178599015041464, 5.68259976238415703207747646114, 6.55407006974261045103044601492, 7.47247250670105126445961996188, 8.379699047398326132676036948695, 8.813587149500236605662161945370, 9.211220947061067274342048917206

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