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Properties

Label 2-2520-280.277-c0-0-2
Degree $2$
Conductor $2520$
Sign $0.882 - 0.470i$
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 + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.500 − 0.866i)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 + 1.00i·25-s + (0.5 + 0.866i)28-s − 1.93·29-s + (0.866 − 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.500 − 0.866i)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 + 1.00i·25-s + (0.5 + 0.866i)28-s − 1.93·29-s + (0.866 − 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.228270452\)
\(L(\frac12)\) \(\approx\) \(2.228270452\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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

−8.911748643719292496480836986791, −8.211525202262267248977555452304, −7.61757193516380658507830255797, −6.88471530341395944335254385979, −5.73251245146626525865532328864, −5.28736906165941083185458196443, −4.27902399369942224770472665458, −3.90757015135040319520667955454, −2.56746262680741381531371555671, −1.54096465013141679325381130876, 1.33221766639219058998625760502, 2.51636608096280979023639368747, 3.55303088362753864100800944566, 4.09927601218610432927925621184, 4.96356818201166045544742664103, 5.83676933291783671906903883164, 6.79231171265596378338603622579, 7.31058275868015242559244046464, 8.035091025110802939005556218700, 8.992348966511355035710149540111

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