Properties

Label 2-2520-840.59-c0-0-5
Degree $2$
Conductor $2520$
Sign $-0.0431 + 0.999i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (−0.866 − 0.499i)10-s + (1.67 + 0.965i)11-s + 0.517i·13-s + (0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s − 0.999·20-s + 1.93·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.258 + 0.448i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (−0.866 − 0.499i)10-s + (1.67 + 0.965i)11-s + 0.517i·13-s + (0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s − 0.999·20-s + 1.93·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.258 + 0.448i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0431 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0431 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0431 + 0.999i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.0431 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.127763475\)
\(L(\frac12)\) \(\approx\) \(2.127763475\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 0.517iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.93T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.105115200751178729275752335472, −8.068865612520966156723464243804, −7.18278394097654887841978287623, −6.65131244222007698433363103554, −5.47616735864394888469979234232, −4.72951787677533256065230129799, −4.09572382895983578918584765421, −3.58450074654919258970586793849, −1.85659532200754494863825006509, −1.27205088101040315961842005909, 1.81947008087954444036104360104, 3.02552869422117859515680005404, 3.66380041736083309172590751153, 4.46993009587217488557714286044, 5.55167865777366586831507883055, 6.21107957952768793325261323976, 6.74560857693519716203142798479, 7.84153303911362572051173017641, 8.204307285978547925781263715826, 9.020651503315034008172141354013

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