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Properties

Label 2-5292-63.59-c1-0-37
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: $\chi_{5292} (2285, \cdot )$
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

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

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