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Properties

Label 2-1380-1380.479-c0-0-0
Degree $2$
Conductor $1380$
Sign $-0.664 - 0.746i$
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  + (−0.755 + 0.654i)2-s + (−0.281 − 0.959i)3-s + (0.142 − 0.989i)4-s + (−0.540 + 0.841i)5-s + (0.841 + 0.540i)6-s + (0.540 + 0.841i)8-s + (−0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.989 + 0.142i)12-s + (0.959 + 0.281i)15-s + (−0.959 − 0.281i)16-s + (−0.989 + 0.857i)17-s + (0.281 − 0.959i)18-s + (−0.186 + 0.215i)19-s + (0.755 + 0.654i)20-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (−0.281 − 0.959i)3-s + (0.142 − 0.989i)4-s + (−0.540 + 0.841i)5-s + (0.841 + 0.540i)6-s + (0.540 + 0.841i)8-s + (−0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.989 + 0.142i)12-s + (0.959 + 0.281i)15-s + (−0.959 − 0.281i)16-s + (−0.989 + 0.857i)17-s + (0.281 − 0.959i)18-s + (−0.186 + 0.215i)19-s + (0.755 + 0.654i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.664 - 0.746i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2934348737\)
\(L(\frac12)\) \(\approx\) \(0.2934348737\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 - 0.654i)T \)
3 \( 1 + (0.281 + 0.959i)T \)
5 \( 1 + (0.540 - 0.841i)T \)
23 \( 1 + (0.909 + 0.415i)T \)
good7 \( 1 + (0.959 - 0.281i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.989 - 0.857i)T + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.186 - 0.215i)T + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.557 - 1.89i)T + (-0.841 - 0.540i)T^{2} \)
37 \( 1 + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - 0.830iT - T^{2} \)
53 \( 1 + (-0.822 + 0.118i)T + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.512 - 1.74i)T + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.474 - 0.304i)T + (0.415 - 0.909i)T^{2} \)
89 \( 1 + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (0.415 + 0.909i)T^{2} \)
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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

−10.23385511841104476938713495990, −8.924951269608606906018561554026, −8.291721608398907349590714443842, −7.62221674880062460583012723573, −6.78959710784464717937084439997, −6.39148278497712170641696158974, −5.47449110096288232286349853681, −4.21773223843714088826424654818, −2.70885714969983418632519668852, −1.61529189689192632706339239609, 0.31485855787382858393530862652, 2.14690117255840829435952699704, 3.46896264861061107935968100505, 4.23356600737199731549825939318, 4.96185147590502576062286026418, 6.14715503526308883110091775968, 7.34383608351398437783857045633, 8.172200319069138987018677691901, 8.933095637247572481235213390692, 9.465526066928380642972640549433

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