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Properties

Label 2-2340-260.227-c0-0-1
Degree $2$
Conductor $2340$
Sign $-0.00863 + 0.999i$
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 + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·8-s + (0.866 − 0.499i)10-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.133i)17-s + 0.999i·20-s + (0.499 + 0.866i)25-s + (0.866 − 0.499i)26-s + (−0.866 − 0.5i)29-s + (−0.499 − 0.866i)32-s + (−0.366 + 0.366i)34-s + (−1.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·8-s + (0.866 − 0.499i)10-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.133i)17-s + 0.999i·20-s + (0.499 + 0.866i)25-s + (0.866 − 0.499i)26-s + (−0.866 − 0.5i)29-s + (−0.499 − 0.866i)32-s + (−0.366 + 0.366i)34-s + (−1.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00863 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00863 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.00863 + 0.999i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.00863 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3472845468\)
\(L(\frac12)\) \(\approx\) \(0.3472845468\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.881458983510503631513239056165, −8.104266001564976172338377989474, −7.48831034912817493805800605692, −6.98355017853281330391557101839, −5.77122444638184464059719122694, −5.20220539340973334318632605744, −4.34510063630086096102587094281, −3.40844477832833191527724739663, −1.81245344969346057681788964419, −0.28402367551332149014345822461, 1.53164347387995186227553287436, 2.77030436131004327929984237913, 3.43771685732487504187831223809, 4.37944164728560165579300009571, 5.12029046824492415092468934739, 6.55906971334438774985289505264, 7.31056210837175164628610218190, 7.88788778513139673720739626064, 8.657601467555114713482251359456, 9.475194544127627533151873365181

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