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Properties

Label 2-2520-280.53-c0-0-2
Degree $2$
Conductor $2520$
Sign $0.417 + 0.908i$
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 + (−0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (0.5 − 0.866i)7-s + (0.707 − 0.707i)8-s + 10-s + (−0.448 − 0.258i)11-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (−0.866 + 0.5i)28-s − 1.93·29-s + (0.866 − 1.5i)31-s + (−0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (0.5 − 0.866i)7-s + (0.707 − 0.707i)8-s + 10-s + (−0.448 − 0.258i)11-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (−0.866 + 0.5i)28-s − 1.93·29-s + (0.866 − 1.5i)31-s + (−0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.417 + 0.908i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.417 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7330809138\)
\(L(\frac12)\) \(\approx\) \(0.7330809138\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + 1.93T + T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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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

−8.796695265597128464161253160691, −7.914698490235703105888645108778, −7.75421209948175331802287948689, −6.77243286937430368307886315135, −5.78061044028881418551528665252, −5.13023135554937320826177203775, −4.35169028813621156168893607046, −3.67252538316539776600065140733, −1.79143990392014471388516378988, −0.53910453418778733373246159854, 1.67966833043243333222177157205, 2.58032464084417568828961235140, 3.28054921971300562809546274879, 4.30243464637435544904120650424, 5.18914317417524081997568950469, 6.03583585067366042605743574209, 7.22107172395795920275465343533, 7.80431602984207507377165079674, 8.606630483332767279743085173989, 9.289711777552553707945236366513

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