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Properties

Label 2-5292-63.5-c1-0-21
Degree $2$
Conductor $5292$
Sign $0.310 + 0.950i$
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  + (−0.842 − 1.45i)5-s + (−3.38 − 1.95i)11-s + (5.24 + 3.02i)13-s + (−0.201 − 0.348i)17-s + (0.145 + 0.0840i)19-s + (−7.69 + 4.44i)23-s + (1.07 − 1.86i)25-s + (6.15 − 3.55i)29-s + 6.28i·31-s + (3.13 − 5.42i)37-s + (1.64 − 2.85i)41-s + (1.80 + 3.12i)43-s + 8.76·47-s + (−4.94 + 2.85i)53-s + 6.58i·55-s + ⋯
L(s)  = 1  + (−0.376 − 0.652i)5-s + (−1.01 − 0.588i)11-s + (1.45 + 0.839i)13-s + (−0.0488 − 0.0845i)17-s + (0.0334 + 0.0192i)19-s + (−1.60 + 0.926i)23-s + (0.215 − 0.373i)25-s + (1.14 − 0.659i)29-s + 1.12i·31-s + (0.514 − 0.891i)37-s + (0.257 − 0.445i)41-s + (0.275 + 0.476i)43-s + 1.27·47-s + (−0.679 + 0.392i)53-s + 0.887i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.310 + 0.950i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.310 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.501329397\)
\(L(\frac12)\) \(\approx\) \(1.501329397\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.842 + 1.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + (7.69 - 4.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.15 + 3.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.28iT - 31T^{2} \)
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 \( 1 - 8.76T + 47T^{2} \)
53 \( 1 + (4.94 - 2.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.50T + 59T^{2} \)
61 \( 1 + 5.12iT - 61T^{2} \)
67 \( 1 + 5.91T + 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (-6.05 + 3.49i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.20T + 79T^{2} \)
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

−8.061407313536073516649685490835, −7.58521714290737349078972879838, −6.42388573154884786700184345180, −5.96514313027971403988569583542, −5.12838988390052028007370334201, −4.27243368209151275299305884654, −3.69648981794702253587397140776, −2.66622489393424882296456503376, −1.58288947316968802854207723469, −0.50165949474639817383062479177, 0.909602362976145479831664344278, 2.26881822534652846920757396692, 2.98327750149063403748204203030, 3.84287415957912588775713849612, 4.57287701478762773850458651454, 5.57871822754864680194919721660, 6.16820269047777257342159682863, 6.91433067235798529780165752891, 7.76511367149086521001368693117, 8.170637747505185925643828692634

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