Properties

Label 2-2520-280.179-c0-0-2
Degree $2$
Conductor $2520$
Sign $-0.832 - 0.553i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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 + (0.5 + 0.866i)5-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)10-s + (−0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·20-s − 0.999·22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)10-s + (−0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·20-s − 0.999·22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.561168475\)
\(L(\frac12)\) \(\approx\) \(1.561168475\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
good11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.461865109041524568028347669085, −8.251572839608973853626951332520, −7.76539118298374296846084543204, −7.11692493458671415712059453737, −6.37139887040370844540446229686, −5.46552772426230396254416443571, −4.86596004918300078878185529404, −4.02596492776142049016658132521, −2.82891471962978275584765359106, −2.02178447027439540955782500175, 0.886730212315747579153929022677, 2.00959894272386569722165383402, 2.82700128728165045926069615453, 4.07851150548218327490554577510, 4.85849633372844742579355715621, 5.40003157953855464434691096290, 6.03372188414248506532996682906, 7.40113234378831909711612688961, 8.179402031079579970020860391638, 9.009573886587004097076194906078

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