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Properties

Label 2-1380-1380.1019-c0-0-1
Degree $2$
Conductor $1380$
Sign $0.635 + 0.771i$
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 + (−0.909 + 0.415i)5-s + (−0.654 + 0.755i)6-s + (0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)10-s + (0.909 + 0.415i)12-s + (0.841 + 0.540i)15-s + (0.415 − 0.909i)16-s + (0.281 + 0.0405i)17-s + (0.989 + 0.142i)18-s + (0.273 + 1.89i)19-s + (0.540 − 0.841i)20-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 + 0.540i)4-s + (−0.909 + 0.415i)5-s + (−0.654 + 0.755i)6-s + (0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)10-s + (0.909 + 0.415i)12-s + (0.841 + 0.540i)15-s + (0.415 − 0.909i)16-s + (0.281 + 0.0405i)17-s + (0.989 + 0.142i)18-s + (0.273 + 1.89i)19-s + (0.540 − 0.841i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.635 + 0.771i$
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: \((2,\ 1380,\ (\ :0),\ 0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5691944976\)
\(L(\frac12)\) \(\approx\) \(0.5691944976\)
\(L(1)\) 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 \( 1 + (0.909 - 0.415i)T \)
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

−9.937439336123001280872130921063, −8.753037791917441182523449130221, −7.958875000373936378000675633866, −7.54127311443040482791781825655, −6.48169018871819946275047997624, −5.44377791158495785400083255965, −4.37151614359186969442927176144, −3.41783181661121038027233531370, −2.37683732662891089545188100087, −1.05833807787673860532738349407, 0.74981324281426474452348118128, 3.24591646334143777466553396373, 4.24859249849273138223160330670, 4.97463943873717710346550188887, 5.55752743255877003827058471594, 6.80408575411055134538712595636, 7.28933644545552521585538109309, 8.455562958826803320180759752655, 8.963418836935571209798236054664, 9.672239130689262438269895501860

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