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Properties

Label 2-2520-280.179-c0-0-4
Degree $2$
Conductor $2520$
Sign $-0.553 + 0.832i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − i·7-s + 0.999·8-s + (0.499 − 0.866i)10-s + (0.866 − 1.5i)11-s − 1.73·13-s + (−0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s − 0.999·20-s − 1.73·22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 1.49i)26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − i·7-s + 0.999·8-s + (0.499 − 0.866i)10-s + (0.866 − 1.5i)11-s − 1.73·13-s + (−0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s − 0.999·20-s − 1.73·22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 1.49i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7890442098\)
\(L(\frac12)\) \(\approx\) \(0.7890442098\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + iT \)
good11 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.73T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.117160375266300867063233172004, −8.155523348984268862702465996476, −7.29328794847616171758112904693, −6.80913440512169974869604992231, −5.78077217044983124208164598929, −4.56466019230544709108542002929, −3.80522931322031377883077033047, −2.89541896766017436989623744916, −2.12153986388494405744184416180, −0.61257256774891144671805280039, 1.59596008749811061438707404629, 2.28155376534316570216692694291, 4.18678632794795845858004239453, 4.85365279820322811340395280069, 5.51940522887001682602665338057, 6.27115650833818183539131750822, 7.15009128331532956246126026346, 7.82133890989389131866393862505, 8.671184150881633977038219605751, 9.346144740525756001768059297217

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