Properties

Label 2-1380-1380.419-c0-0-3
Degree $2$
Conductor $1380$
Sign $0.525 - 0.850i$
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  + (−0.281 + 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (0.755 − 0.345i)7-s + (0.755 − 0.654i)8-s + (−0.415 + 0.909i)9-s + (−0.909 − 0.415i)10-s + i·12-s + (0.118 + 0.822i)14-s + (0.909 − 0.415i)15-s + (0.415 + 0.909i)16-s + (−0.755 − 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (0.755 − 0.345i)7-s + (0.755 − 0.654i)8-s + (−0.415 + 0.909i)9-s + (−0.909 − 0.415i)10-s + i·12-s + (0.118 + 0.822i)14-s + (0.909 − 0.415i)15-s + (0.415 + 0.909i)16-s + (−0.755 − 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7532333758\)
\(L(\frac12)\) \(\approx\) \(0.7532333758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 - 0.959i)T \)
3 \( 1 + (0.540 + 0.841i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-0.281 + 0.959i)T \)
good7 \( 1 + (-0.755 + 0.345i)T + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + 0.284iT - T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.215 - 1.49i)T + (-0.959 + 0.281i)T^{2} \)
89 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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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

−9.949078831209582084534239976945, −8.777215211735996060546952757889, −7.995562293233147248801159578137, −7.39459654074257212300654976158, −6.72898936790547466825239453777, −6.09023246827539512262796779761, −5.11173683206748492004367553958, −4.24413900121558454659755122206, −2.67783099432642151200313360046, −1.19382935828985861458980881642, 0.953437054036306454062667306169, 2.37974252211667679774054860327, 3.77239047338406427032562878785, 4.46241588110557540634403244436, 5.17350092850270335205640183131, 5.94204495992401865600556186412, 7.60362716944515432020738387773, 8.334588919066382693863856152829, 9.133441087743817187706884942747, 9.570744882370265738652323758004

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