Properties

Label 2-2520-2520.2477-c0-0-5
Degree $2$
Conductor $2520$
Sign $0.488 + 0.872i$
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 + (−0.923 − 0.382i)3-s + (−0.866 + 0.499i)4-s + (0.382 + 0.923i)5-s + (0.130 − 0.991i)6-s + (−0.965 + 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.793 + 0.608i)10-s + (0.991 − 0.130i)12-s + (−1.78 − 0.478i)13-s + (−0.499 − 0.866i)14-s i·15-s + (0.500 − 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.98i·19-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.923 − 0.382i)3-s + (−0.866 + 0.499i)4-s + (0.382 + 0.923i)5-s + (0.130 − 0.991i)6-s + (−0.965 + 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.793 + 0.608i)10-s + (0.991 − 0.130i)12-s + (−1.78 − 0.478i)13-s + (−0.499 − 0.866i)14-s i·15-s + (0.500 − 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.98i·19-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1973801757\)
\(L(\frac12)\) \(\approx\) \(0.1973801757\)
\(L(1)\) 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 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.98iT - T^{2} \)
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 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \)
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

−9.055376266613086082306032772442, −7.69981684942707339917277403185, −7.19696841320687403972246817022, −6.69638698261669560540752933738, −6.00256701925349757212612997990, −5.25297035280203025903031340448, −4.58208316111702307435692483925, −3.21161047986802531808366342439, −2.42641475214111806477195217072, −0.13794078230520995569868825107, 1.32136864529541315252781637620, 2.50379581389533882633984901121, 3.81202261569343824926769015892, 4.37680727519194360417275841981, 5.22871175540703511941269224512, 5.82208412312819871670619686871, 6.63946011966482363681948772805, 7.75806669427142724339789162185, 8.877187239561338593964319225134, 9.592982217258234022638893281418

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