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Properties

Label 2-3040-760.619-c0-0-0
Degree $2$
Conductor $3040$
Sign $0.671 + 0.740i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (1 + 1.73i)13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)35-s + 37-s + (0.5 − 0.866i)41-s − 0.999·45-s + (−1 − 1.73i)47-s + (−0.5 − 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (1 + 1.73i)13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)35-s + 37-s + (0.5 − 0.866i)41-s − 0.999·45-s + (−1 − 1.73i)47-s + (−0.5 − 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297371748\)
\(L(\frac12)\) \(\approx\) \(1.297371748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.000732084135495242370467687617, −8.377116103663000337426178673480, −6.93721810886196574099493167579, −6.56065178387043791894861137201, −5.91933392737660480258898665736, −4.96400710779247508866763562378, −3.92775369384416416270841555023, −3.41712815029325000519048335661, −2.02194835953953711447701553187, −0.937359203617292703491779511416, 1.29828814189414780789059437273, 2.83135266384425356451791987173, 3.09414581716443576160011863329, 4.17343833389616459285210140999, 5.45966493862891719914594698633, 6.07407167079660713860752937162, 6.49728019716131598792827613114, 7.58042657672888258654992696404, 8.142584297363673569685049422218, 9.085091548687465994327044850140

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