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Properties

Label 2-5292-63.59-c1-0-8
Degree $2$
Conductor $5292$
Sign $-0.827 - 0.562i$
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  + 0.0764·5-s + 5.38i·11-s + (4.60 + 2.65i)13-s + (−1.89 + 3.27i)17-s + (4.33 − 2.50i)19-s + 2.33i·23-s − 4.99·25-s + (−8.84 + 5.10i)29-s + (−4.97 + 2.87i)31-s + (0.354 + 0.613i)37-s + (3.29 − 5.71i)41-s + (0.716 + 1.24i)43-s + (−1.46 + 2.53i)47-s + (−10.4 − 6.05i)53-s + 0.411i·55-s + ⋯
L(s)  = 1  + 0.0341·5-s + 1.62i·11-s + (1.27 + 0.737i)13-s + (−0.458 + 0.794i)17-s + (0.995 − 0.574i)19-s + 0.487i·23-s − 0.998·25-s + (−1.64 + 0.948i)29-s + (−0.893 + 0.516i)31-s + (0.0582 + 0.100i)37-s + (0.515 − 0.892i)41-s + (0.109 + 0.189i)43-s + (−0.213 + 0.369i)47-s + (−1.44 − 0.831i)53-s + 0.0554i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.827 - 0.562i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.827 - 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.175019019\)
\(L(\frac12)\) \(\approx\) \(1.175019019\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.0764T + 5T^{2} \)
11 \( 1 - 5.38iT - 11T^{2} \)
13 \( 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.89 - 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.33 + 2.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.33iT - 23T^{2} \)
29 \( 1 + (8.84 - 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.354 - 0.613i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.29 + 5.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.716 - 1.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.46 - 2.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.4 + 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.289 + 0.502i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.40 - 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.63 + 4.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.32iT - 71T^{2} \)
73 \( 1 + (-6.17 - 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.469 - 0.812i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.49 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.51 + 2.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.18 - 3.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.555766786378129487542414366684, −7.55790192560232362731514855055, −7.18764757746605185563282928889, −6.36547650252168320509944346501, −5.60199765040787179534138021784, −4.80909969725965630704984498238, −3.97913058829842787392150343228, −3.38959561653394585902775666418, −1.97226771500967756503373466578, −1.55449456162384735209462609721, 0.30711599072691271137574736038, 1.32403285945678071695111194215, 2.57202407377878418198170846704, 3.48233264572825387147595725802, 3.92305852948332194401132331175, 5.17077685077169775796781530140, 5.95864385693060038453443318527, 6.08890424144124219780425070068, 7.36405838061869091860387575451, 7.934760826136128835151267243822

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