Properties

Label 2-2520-2520.797-c0-0-5
Degree $2$
Conductor $2520$
Sign $0.982 + 0.187i$
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  + (0.965 + 0.258i)2-s + (0.382 − 0.923i)3-s + (0.866 + 0.499i)4-s + (0.923 − 0.382i)5-s + (0.608 − 0.793i)6-s + (−0.258 + 0.965i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.991 − 0.130i)10-s + (0.793 − 0.608i)12-s + (0.198 + 0.739i)13-s + (−0.499 + 0.866i)14-s i·15-s + (0.500 + 0.866i)16-s + (−0.5 − 0.866i)18-s − 1.58i·19-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.382 − 0.923i)3-s + (0.866 + 0.499i)4-s + (0.923 − 0.382i)5-s + (0.608 − 0.793i)6-s + (−0.258 + 0.965i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.991 − 0.130i)10-s + (0.793 − 0.608i)12-s + (0.198 + 0.739i)13-s + (−0.499 + 0.866i)14-s i·15-s + (0.500 + 0.866i)16-s + (−0.5 − 0.866i)18-s − 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.759033012\)
\(L(\frac12)\) \(\approx\) \(2.759033012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.58iT - T^{2} \)
23 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
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

−8.988964743328573833239708846650, −8.240959921676509247432576684700, −7.37450574839785687753055005500, −6.46276845931942396007138417937, −6.11054977914459474519098420160, −5.34852498017430185948816755015, −4.40995438251819002013899484879, −3.18384109081006450428154271446, −2.34016526370575843952182991497, −1.72708871212233046978396059679, 1.67487444112240924189065883340, 2.70391841363885151906762131545, 3.60432588948813795809454946676, 4.11142847423277025343156253657, 5.15582339756123122295707219201, 5.90604759657361563603167798337, 6.44788451374625227211039363537, 7.60845751608285594422377210766, 8.234068492569865984491234665058, 9.599973364413482373694118686352

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