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Properties

Label 2-5292-63.59-c1-0-24
Degree $2$
Conductor $5292$
Sign $0.999 + 0.0292i$
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  + 0.699·5-s + 0.265i·11-s + (1.13 + 0.657i)13-s + (1.86 − 3.22i)17-s + (0.382 − 0.220i)19-s + 4.96i·23-s − 4.51·25-s + (0.273 − 0.157i)29-s + (4.85 − 2.80i)31-s + (−0.351 − 0.608i)37-s + (5.39 − 9.34i)41-s + (3.73 + 6.46i)43-s + (−3.50 + 6.06i)47-s + (8.51 + 4.91i)53-s + 0.185i·55-s + ⋯
L(s)  = 1  + 0.312·5-s + 0.0799i·11-s + (0.315 + 0.182i)13-s + (0.452 − 0.783i)17-s + (0.0877 − 0.0506i)19-s + 1.03i·23-s − 0.902·25-s + (0.0507 − 0.0292i)29-s + (0.872 − 0.503i)31-s + (−0.0577 − 0.0999i)37-s + (0.842 − 1.45i)41-s + (0.569 + 0.985i)43-s + (−0.510 + 0.884i)47-s + (1.17 + 0.675i)53-s + 0.0250i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.999 + 0.0292i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.999 + 0.0292i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.124624571\)
\(L(\frac12)\) \(\approx\) \(2.124624571\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.699T + 5T^{2} \)
11 \( 1 - 0.265iT - 11T^{2} \)
13 \( 1 + (-1.13 - 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.86 + 3.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.382 + 0.220i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (-0.273 + 0.157i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.85 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.39 + 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.50 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.51 - 4.91i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.89 - 2.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.97 - 5.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (-6.66 - 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.698 - 1.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.72 - 6.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.59 + 9.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.012707334504963324878366755562, −7.58825337084056516307312662952, −6.75133969743058215868509444682, −5.94982954087198942473829606202, −5.41028290813681782797774951898, −4.49694132722452759002058364175, −3.70990213342809434953567145185, −2.80358388811510321820294952068, −1.89741470723225767147183837964, −0.792387376471995345160151300046, 0.809162717040252515480701007743, 1.89712124927151622917344450683, 2.81643078229869747107203754839, 3.73491203085917040513961403018, 4.47411242261938658074555349155, 5.40684412666817898245787926747, 6.04475451910804454965555638086, 6.66303884329664363790901626910, 7.52820636842487378659566979637, 8.318161715970241131763463429485

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