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Properties

Label 2-1380-1380.1019-c0-0-4
Degree $2$
Conductor $1380$
Sign $0.167 - 0.985i$
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 + (−0.909 + 0.415i)5-s + (0.415 + 0.909i)6-s + (0.909 + 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.281 + 0.959i)12-s + (−0.841 − 0.540i)15-s + (0.841 + 0.540i)16-s + (0.281 + 0.0405i)17-s + (−0.540 + 0.841i)18-s + (−0.273 − 1.89i)19-s + (−0.989 + 0.142i)20-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (0.540 + 0.841i)3-s + (0.959 + 0.281i)4-s + (−0.909 + 0.415i)5-s + (0.415 + 0.909i)6-s + (0.909 + 0.415i)8-s + (−0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.281 + 0.959i)12-s + (−0.841 − 0.540i)15-s + (0.841 + 0.540i)16-s + (0.281 + 0.0405i)17-s + (−0.540 + 0.841i)18-s + (−0.273 − 1.89i)19-s + (−0.989 + 0.142i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.167 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.030646104\)
\(L(\frac12)\) \(\approx\) \(2.030646104\)
\(L(1)\) 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 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (0.755 - 0.654i)T \)
good7 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (-0.281 - 0.0405i)T + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + 1.30iT - T^{2} \)
53 \( 1 + (-0.368 - 1.25i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.817 + 1.27i)T + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.654 + 0.755i)T^{2} \)
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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

−10.17807757623950440277092197181, −9.046396180135629647022315482518, −8.247500389935699753418520095265, −7.43347829884058256468364370216, −6.76232407831201946504424686152, −5.54408232582525919374232091791, −4.71927822024548779809708490035, −3.95660611069385738276536960194, −3.21500449271784700723824744500, −2.32104314393840448169864628377, 1.34801282491056077057557004788, 2.52298442266482731819038409573, 3.66376984853158855794504379676, 4.17909051465652008782570085354, 5.49737961892744990420414226147, 6.24026392732171305920909249087, 7.19097994496120960377701154357, 7.903124160929079454592249712006, 8.418937188265458327875525317868, 9.626853678017722948747173086951

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