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Properties

Label 2-1380-1380.1079-c0-0-6
Degree $2$
Conductor $1380$
Sign $0.938 + 0.343i$
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 + (0.959 + 0.281i)5-s + (0.142 + 0.989i)6-s + (−0.989 − 0.857i)7-s + (0.989 + 0.142i)8-s + (0.142 − 0.989i)9-s + (−0.755 + 0.654i)10-s + (−0.909 − 0.415i)12-s + (1.25 − 0.368i)14-s + (0.909 − 0.415i)15-s + (−0.654 + 0.755i)16-s + (0.755 + 0.654i)18-s + (−0.142 − 0.989i)20-s − 1.30·21-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (0.755 − 0.654i)3-s + (−0.415 − 0.909i)4-s + (0.959 + 0.281i)5-s + (0.142 + 0.989i)6-s + (−0.989 − 0.857i)7-s + (0.989 + 0.142i)8-s + (0.142 − 0.989i)9-s + (−0.755 + 0.654i)10-s + (−0.909 − 0.415i)12-s + (1.25 − 0.368i)14-s + (0.909 − 0.415i)15-s + (−0.654 + 0.755i)16-s + (0.755 + 0.654i)18-s + (−0.142 − 0.989i)20-s − 1.30·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.938 + 0.343i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.938 + 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088688260\)
\(L(\frac12)\) \(\approx\) \(1.088688260\)
\(L(1)\) 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 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
good7 \( 1 + (0.989 + 0.857i)T + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.822 - 0.118i)T + (0.959 - 0.281i)T^{2} \)
47 \( 1 - 1.91iT - T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (-0.153 + 0.239i)T + (-0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \)
89 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 + 0.540i)T^{2} \)
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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

−9.367841609876615725661499611342, −9.100284070238348258701683772896, −7.997473860084657981245435806072, −7.23385174625470883837638977438, −6.54192022695122219674636402077, −6.14316544648063971909977075056, −4.83964415461160301191943500444, −3.55768617720367207807267158484, −2.41773781970468026112220186705, −1.08821658221527670845838023113, 1.74860844966218088532665875776, 2.73944920270315684466418109267, 3.35408296647491932284791448461, 4.59331754448904901064679394692, 5.48535650737451546581997116197, 6.59251095243977096239248717659, 7.74829409142498000129524794756, 8.749114473424176277090137150831, 9.075898963617621227093470734627, 9.867575233282952932476829144286

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