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Properties

Label 2-2520-2520.2477-c0-0-3
Degree $2$
Conductor $2520$
Sign $-0.0136 + 0.999i$
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  + (−0.258 − 0.965i)2-s + (−0.382 − 0.923i)3-s + (−0.866 + 0.499i)4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.130 + 0.991i)10-s + (0.793 + 0.608i)12-s + (1.78 + 0.478i)13-s + (0.499 + 0.866i)14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s − 1.21i·19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.382 − 0.923i)3-s + (−0.866 + 0.499i)4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.130 + 0.991i)10-s + (0.793 + 0.608i)12-s + (1.78 + 0.478i)13-s + (0.499 + 0.866i)14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s − 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0136 + 0.999i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.0136 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5783485311\)
\(L(\frac12)\) \(\approx\) \(0.5783485311\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.21iT - T^{2} \)
23 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.793 + 1.37i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−8.903287490212388380681590149295, −8.258769839440015503101687933763, −7.49455064213313763004489593468, −6.70444988136390938911195471725, −5.84712743320767800443947595174, −4.85097595113090488894646136321, −3.69688172125175613250799598663, −3.21996817578532277176595914503, −1.94333846474339680527151085424, −0.78419546430315169502782000294, 0.73256605985713212924672187335, 3.20329923843205659276944978048, 3.91698780365689999440888362944, 4.40176075454693580079574903700, 5.64593637607992115662820334593, 6.20427401976611485610364991021, 6.81081688828747081981406739023, 7.912495230600610778596344762357, 8.526696751430362994412894998006, 9.041703626739663900274692860129

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