Properties

Label 2-7920-33.32-c1-0-36
Degree $2$
Conductor $7920$
Sign $-0.812 - 0.583i$
Analytic cond. $63.2415$
Root an. cond. $7.95245$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + 1.15i·7-s + (1.08 + 3.13i)11-s + 6.48i·13-s − 0.176·17-s + 5.77i·19-s + 7.84i·23-s − 25-s + 9.47·29-s + 0.163·31-s − 1.15·35-s + 3.01·37-s + 2.86·41-s − 5.38i·43-s − 2.23i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.436i·7-s + (0.326 + 0.945i)11-s + 1.79i·13-s − 0.0427·17-s + 1.32i·19-s + 1.63i·23-s − 0.200·25-s + 1.75·29-s + 0.0293·31-s − 0.195·35-s + 0.496·37-s + 0.447·41-s − 0.820i·43-s − 0.325i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.812 - 0.583i$
Analytic conductor: \(63.2415\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7920} (3761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7920,\ (\ :1/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.982366728\)
\(L(\frac12)\) \(\approx\) \(1.982366728\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - iT \)
11 \( 1 + (-1.08 - 3.13i)T \)
good7 \( 1 - 1.15iT - 7T^{2} \)
13 \( 1 - 6.48iT - 13T^{2} \)
17 \( 1 + 0.176T + 17T^{2} \)
19 \( 1 - 5.77iT - 19T^{2} \)
23 \( 1 - 7.84iT - 23T^{2} \)
29 \( 1 - 9.47T + 29T^{2} \)
31 \( 1 - 0.163T + 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 - 2.86T + 41T^{2} \)
43 \( 1 + 5.38iT - 43T^{2} \)
47 \( 1 + 2.23iT - 47T^{2} \)
53 \( 1 + 3.01iT - 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 + 6.46T + 67T^{2} \)
71 \( 1 - 8.28iT - 71T^{2} \)
73 \( 1 + 16.0iT - 73T^{2} \)
79 \( 1 + 6.76iT - 79T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.021828091881970479148129417726, −7.32538429549755070022854542869, −6.79751176509586541784609920215, −6.10562085811350313208293094322, −5.42405888425601409802668823965, −4.38357424665067872752956260812, −4.01364431502218433615711600368, −2.97549784767547127312552191053, −2.03722267107118002646569408940, −1.42988248827003305352967862106, 0.57270086355851070776647180696, 0.947543367187198301093991789293, 2.59170841222921971489920792165, 3.02764007461777971600496126007, 4.06315748974048470848639844074, 4.78389191530063879635277789917, 5.39122495403140975141562911279, 6.30731853776420475380776905486, 6.69754108082986071378381110092, 7.80232777379590012183895281520

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