Properties

Label 2-2520-2520.2029-c0-0-0
Degree $2$
Conductor $2520$
Sign $-0.422 - 0.906i$
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  + (−0.866 + 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.965 − 0.258i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + 1.00i·9-s + (0.258 − 0.965i)10-s + (0.965 − 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.93·19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.965 − 0.258i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + 1.00i·9-s + (0.258 − 0.965i)10-s + (0.965 − 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.93·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (2029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001982685\)
\(L(\frac12)\) \(\approx\) \(1.001982685\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.93T + T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
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 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−9.060827801694626625253217697607, −8.518152216688921670251984171300, −8.056061082243347406341869675034, −7.16245831403649169969701411895, −6.63757282720938611942287970890, −5.48141947192830191665642279098, −4.43583001645240909543863249717, −3.83119141101972613898093414660, −2.62288865317525875937267178256, −1.53502837095954838718684145697, 0.885266853027334469366328742467, 1.84063468503155436577055106439, 2.83635396406885614419112803069, 3.79322879956393887993375213166, 4.64146840707818720527583581705, 5.98183901851900435244832990985, 6.85113408986696752919164470525, 7.70953856097570732273700320218, 8.391542761708705346115184144759, 8.648755424438601491229444783724

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