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Properties

Label 2-2520-2520.2477-c0-0-6
Degree $2$
Conductor $2520$
Sign $-0.872 + 0.488i$
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 + (0.923 − 0.382i)5-s + (−0.991 − 0.130i)6-s + (0.965 − 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.608 − 0.793i)10-s + (0.130 + 0.991i)12-s + (−0.739 − 0.198i)13-s + (−0.499 − 0.866i)14-s − i·15-s + (0.500 − 0.866i)16-s + (−0.500 + 0.866i)18-s − 0.261i·19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.382 − 0.923i)3-s + (−0.866 + 0.499i)4-s + (0.923 − 0.382i)5-s + (−0.991 − 0.130i)6-s + (0.965 − 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.608 − 0.793i)10-s + (0.130 + 0.991i)12-s + (−0.739 − 0.198i)13-s + (−0.499 − 0.866i)14-s − i·15-s + (0.500 − 0.866i)16-s + (−0.500 + 0.866i)18-s − 0.261i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.872 + 0.488i$
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: \((2,\ 2520,\ (\ :0),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351442559\)
\(L(\frac12)\) \(\approx\) \(1.351442559\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 0.261iT - T^{2} \)
23 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.93iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
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

−8.807742014900555704214415844812, −8.135531633106920123957021590387, −7.54051956278118795165594312151, −6.59051609372724556675668913381, −5.42716217927190740969663927550, −4.87032268528021594844867654686, −3.70105758441735814100833043191, −2.54665563458438587872163461507, −1.92433807706978497783728339357, −1.00199677564371279151703739797, 1.75970425244338328518855641151, 2.81140264592599784594495439027, 4.14777515784621050456929216350, 4.85498725397951444350569528765, 5.51121545393202469328079774568, 6.20436740442342177748604190885, 7.26253643058707756012440530732, 7.914235559251743760059699927344, 8.819433165648113335980353852172, 9.166825595963461532319653158359

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