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Properties

Label 2-1380-1380.1019-c0-0-6
Degree $2$
Conductor $1380$
Sign $0.305 - 0.952i$
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 + (0.989 + 0.142i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.654 + 0.755i)6-s + (−0.540 − 1.84i)7-s + (−0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (−0.281 + 0.959i)10-s + i·12-s + (0.797 − 1.74i)14-s + (0.281 + 0.959i)15-s + (−0.959 + 0.281i)16-s + (0.540 + 0.841i)18-s + (−0.841 + 0.540i)20-s + (−0.273 − 1.89i)21-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (0.989 + 0.142i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.654 + 0.755i)6-s + (−0.540 − 1.84i)7-s + (−0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (−0.281 + 0.959i)10-s + i·12-s + (0.797 − 1.74i)14-s + (0.281 + 0.959i)15-s + (−0.959 + 0.281i)16-s + (0.540 + 0.841i)18-s + (−0.841 + 0.540i)20-s + (−0.273 − 1.89i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.305 - 0.952i$
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: \((2,\ 1380,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.187008129\)
\(L(\frac12)\) \(\approx\) \(2.187008129\)
\(L(1)\) 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 \( 1 + (-0.989 - 0.142i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
good7 \( 1 + (0.540 + 1.84i)T + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.153 - 0.239i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - 0.830iT - T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-1.07 + 1.66i)T + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
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

−9.895658960379867433574219884289, −9.156327083855604275473594392235, −7.84917943218269135527930897850, −7.50721685396718697667154091004, −6.76640078006879815494490953820, −6.05514410199729875207192049960, −4.62990213450653099150374670828, −3.80148400016844590251096677712, −3.30240967386653121305884798490, −2.11138559480286670070865566585, 1.72407374219126442430877217525, 2.39032491656742783124378799833, 3.37566834303367872777668747113, 4.37057023543079504499623763167, 5.57216272073395317861143468364, 5.83836174272644682274040903789, 7.09653185328274364956604748921, 8.364567638919904610592227042283, 9.016097291738681225421719978771, 9.511341444307044508458125795630

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