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Properties

Label 2-5292-63.47-c1-0-2
Degree $2$
Conductor $5292$
Sign $-0.962 + 0.269i$
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  − 2.96·5-s + 4.72i·11-s + (−3.54 + 2.04i)13-s + (0.835 + 1.44i)17-s + (4.25 + 2.45i)19-s + 4.91i·23-s + 3.82·25-s + (−0.238 − 0.137i)29-s + (1.38 + 0.801i)31-s + (−1.69 + 2.93i)37-s + (3.55 + 6.15i)41-s + (5.22 − 9.05i)43-s + (−5.49 − 9.52i)47-s + (0.707 − 0.408i)53-s − 14.0i·55-s + ⋯
L(s)  = 1  − 1.32·5-s + 1.42i·11-s + (−0.981 + 0.566i)13-s + (0.202 + 0.350i)17-s + (0.975 + 0.563i)19-s + 1.02i·23-s + 0.764·25-s + (−0.0442 − 0.0255i)29-s + (0.249 + 0.143i)31-s + (−0.278 + 0.483i)37-s + (0.555 + 0.961i)41-s + (0.797 − 1.38i)43-s + (−0.802 − 1.38i)47-s + (0.0971 − 0.0560i)53-s − 1.89i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.962 + 0.269i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4308736905\)
\(L(\frac12)\) \(\approx\) \(0.4308736905\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.96T + 5T^{2} \)
11 \( 1 - 4.72iT - 11T^{2} \)
13 \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.835 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 + (0.238 + 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 0.801i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.49 + 9.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.707 + 0.408i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.37 + 2.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.23 + 3.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (13.6 - 7.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.15 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.60 - 7.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.00 - 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.385848871947739068920451792230, −7.72388739763614710657375081747, −7.27100724015352413549247073428, −6.75021749909425772480487540773, −5.51185406611829556154616631881, −4.85504115670847278134190541392, −4.09769344974167335084381207595, −3.51916815986788260406671735545, −2.41871053654543240266468545183, −1.39524685407703344311576079071, 0.14556926157201245617370326428, 0.932907208414183976789599319726, 2.70630791573099844443530694382, 3.14103216284432605842217350001, 4.07796115197023188095524903784, 4.79828251994526108935446763424, 5.59054666207049040567890291938, 6.38703290085790585909636576100, 7.41458255992244402629691539389, 7.66848560826546960531518480121

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