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Properties

Label 2-5292-21.20-c1-0-4
Degree $2$
Conductor $5292$
Sign $-0.654 - 0.755i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  − 5.19i·13-s + 3.46i·19-s − 5·25-s + 8.66i·31-s − 11·37-s − 5·43-s − 8.66i·61-s + 5·67-s + 13.8i·73-s + 17·79-s + 19.0i·97-s + 15.5i·103-s − 17·109-s + ⋯
L(s)  = 1  − 1.44i·13-s + 0.794i·19-s − 25-s + 1.55i·31-s − 1.80·37-s − 0.762·43-s − 1.10i·61-s + 0.610·67-s + 1.62i·73-s + 1.91·79-s + 1.93i·97-s + 1.53i·103-s − 1.62·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5930767204\)
\(L(\frac12)\) \(\approx\) \(0.5930767204\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−8.241001694185974995527812897365, −7.936869188063314151670597058975, −7.00310059293055954548824579682, −6.33140145751462801583483214359, −5.38983429827543614124661030438, −5.07339055811142754661614804588, −3.77312625595064295626551909776, −3.32246369656432089808295845327, −2.23290440433770820271489731249, −1.19358987372700511477429586062, 0.15649257901959477186067883919, 1.64643302784307547116721143810, 2.36342405152180623731886687414, 3.50427762351055442990088046558, 4.22353444123110275020389250435, 4.94404416408286246665793372854, 5.81822557783620274464483646378, 6.57858043849105130473817358519, 7.14996074477411638312342848278, 7.906443356470116545521637909581

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