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Properties

Label 2-5292-63.59-c1-0-12
Degree $2$
Conductor $5292$
Sign $0.756 - 0.653i$
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  − 1.68·5-s + 3.90i·11-s + (−5.24 − 3.02i)13-s + (0.201 − 0.348i)17-s + (0.145 − 0.0840i)19-s − 8.88i·23-s − 2.15·25-s + (6.15 − 3.55i)29-s + (−5.44 + 3.14i)31-s + (3.13 + 5.42i)37-s + (−1.64 + 2.85i)41-s + (1.80 + 3.12i)43-s + (4.38 − 7.59i)47-s + (4.94 + 2.85i)53-s − 6.58i·55-s + ⋯
L(s)  = 1  − 0.753·5-s + 1.17i·11-s + (−1.45 − 0.839i)13-s + (0.0488 − 0.0845i)17-s + (0.0334 − 0.0192i)19-s − 1.85i·23-s − 0.431·25-s + (1.14 − 0.659i)29-s + (−0.977 + 0.564i)31-s + (0.514 + 0.891i)37-s + (−0.257 + 0.445i)41-s + (0.275 + 0.476i)43-s + (0.639 − 1.10i)47-s + (0.679 + 0.392i)53-s − 0.887i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.756 - 0.653i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.756 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.079184453\)
\(L(\frac12)\) \(\approx\) \(1.079184453\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.68T + 5T^{2} \)
11 \( 1 - 3.90iT - 11T^{2} \)
13 \( 1 + (5.24 + 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.145 + 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.88iT - 23T^{2} \)
29 \( 1 + (-6.15 + 3.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.44 - 3.14i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.80 - 3.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.38 + 7.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.94 - 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.25 - 3.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.43 + 2.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.95 - 5.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (-6.05 - 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.603 - 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.181 + 0.314i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.38 - 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.508 - 0.293i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.103970265145075407528935499603, −7.59616623133027606836622565449, −6.98190352176965223620011217557, −6.24823689992348143190750756539, −5.09946764266122086470599524953, −4.67707844109641919603748467227, −3.92260322012577226335534291468, −2.80669795795517919416863689755, −2.20685052600403456946550856069, −0.68820178676888334508772464950, 0.43275791859921606025046852021, 1.77965645095662397324478523377, 2.82601938134369198790927867397, 3.68024235923881465493271781283, 4.28777782200910115379908816857, 5.28760904152803126974735403605, 5.82248634296423849604756311240, 6.87198936674749406165331639513, 7.45067714065404896616310664036, 7.966227886540949747842508761022

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