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Properties

Label 2-2520-2520.1973-c0-0-4
Degree $2$
Conductor $2520$
Sign $0.944 + 0.329i$
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 + (0.793 − 0.608i)5-s + (−0.991 − 0.130i)6-s + (−0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.608 + 0.793i)10-s + (0.991 − 0.130i)12-s + (0.198 − 0.739i)13-s + (0.499 + 0.866i)14-s + (0.965 − 0.258i)15-s + (0.500 − 0.866i)16-s + (−0.866 − 0.500i)18-s + 0.261i·19-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.923 + 0.382i)3-s + (0.866 − 0.499i)4-s + (0.793 − 0.608i)5-s + (−0.991 − 0.130i)6-s + (−0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.608 + 0.793i)10-s + (0.991 − 0.130i)12-s + (0.198 − 0.739i)13-s + (0.499 + 0.866i)14-s + (0.965 − 0.258i)15-s + (0.500 − 0.866i)16-s + (−0.866 − 0.500i)18-s + 0.261i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.944 + 0.329i$
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: \((2,\ 2520,\ (\ :0),\ 0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246696989\)
\(L(\frac12)\) \(\approx\) \(1.246696989\)
\(L(1)\) 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 \( 1 + (-0.793 + 0.608i)T \)
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 \( 1 - 0.261iT - T^{2} \)
23 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \)
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

−8.982922630883365864612975010827, −8.420600763869975605852408378764, −7.74268853036636066992207998786, −7.01348616487407295465511491790, −6.08826197305793450121732807969, −5.24065307311128666098042894489, −4.21092061789409564920335741337, −3.15296059987007250158751432014, −2.14025260340513366568880146910, −1.09579773635875507216661769321, 1.58059209407329048511923391076, 2.32189655747850114278241740393, 2.97197266176623388148242606527, 3.93485104069042891956655372488, 5.53304845650293512635692551665, 6.46040977960527385038500892835, 6.85545679399299817940010868579, 7.77980309576997338778144617380, 8.583069775499509414341760915837, 9.123923240663222239698657459520

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