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Properties

Label 2-2340-2340.1039-c0-0-5
Degree $2$
Conductor $2340$
Sign $-0.766 + 0.642i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
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  + (−0.5 − 0.866i)2-s − i·3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)6-s + 0.999·8-s − 9-s − 0.999·10-s + (0.866 + 1.5i)11-s + (0.866 + 0.499i)12-s + (0.5 − 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + 1.73·19-s + (0.499 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − i·3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)6-s + 0.999·8-s − 9-s − 0.999·10-s + (0.866 + 1.5i)11-s + (0.866 + 0.499i)12-s + (0.5 − 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + 1.73·19-s + (0.499 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023154521\)
\(L(\frac12)\) \(\approx\) \(1.023154521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + iT \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.137666756607444907423938147800, −7.968943929789013885410091199438, −7.73724338134141498342244928020, −6.70725474757530813947660337775, −5.71553584976334165583652810264, −4.84979821705487008294227435349, −3.84512617205539265200951578821, −2.68216177727182389958648606409, −1.73335580312741655932349145265, −0.978423838209223911295764501775, 1.43151840045065731886016545629, 3.16854550771410183984683091499, 3.72554453358336893932114731584, 5.02479758508331546814592854695, 5.63185820681886494487051998006, 6.44480274407306018587053737754, 6.94154877164749361744479568005, 8.119828887448112418826734248262, 8.850370101767836838211275366803, 9.349382502603466700540368976596

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