Properties

Label 2-2520-2520.1973-c0-0-1
Degree $2$
Conductor $2520$
Sign $-0.187 - 0.982i$
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 + (0.923 − 0.382i)3-s + (0.866 − 0.499i)4-s + (−0.382 + 0.923i)5-s + (−0.793 + 0.608i)6-s + (0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.130 − 0.991i)10-s + (0.608 − 0.793i)12-s + (−0.478 + 1.78i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.21i·19-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.923 − 0.382i)3-s + (0.866 − 0.499i)4-s + (−0.382 + 0.923i)5-s + (−0.793 + 0.608i)6-s + (0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.130 − 0.991i)10-s + (0.608 − 0.793i)12-s + (−0.478 + 1.78i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.21i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9022653273\)
\(L(\frac12)\) \(\approx\) \(0.9022653273\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 1.21iT - T^{2} \)
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 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.255413009970473077531949231762, −8.377056035389600240216522009377, −7.979154575521208145703614109322, −7.17375098781428113771384522187, −6.48625100534317922264659178742, −5.90075112129201158524856949019, −4.38406118817166593723343001283, −3.39032927909670603293622582854, −2.18394910971531325437206815374, −1.95276821965764343499660290704, 0.69847549036653746355291183435, 1.95874048723250395248353541971, 3.07246450541036422496957855148, 3.87078357187612953655538016652, 4.70859183282133540277261758520, 5.73508813926161005244589581562, 7.18004122766795120399828868107, 7.62248252056786100134855633790, 8.254639759188132589060139596622, 8.712579853832317743904472596941

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