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Properties

Label 2-5292-63.4-c1-0-8
Degree $2$
Conductor $5292$
Sign $0.0788 - 0.996i$
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  + 3·5-s − 3·11-s + (−0.5 + 0.866i)13-s + (−3 + 5.19i)17-s + (−2 − 3.46i)19-s + 3·23-s + 4·25-s + (1.5 + 2.59i)29-s + (2.5 + 4.33i)31-s + (−1 − 1.73i)37-s + (−1.5 + 2.59i)41-s + (0.5 + 0.866i)43-s + (4.5 − 7.79i)47-s + (−3 + 5.19i)53-s − 9·55-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.904·11-s + (−0.138 + 0.240i)13-s + (−0.727 + 1.26i)17-s + (−0.458 − 0.794i)19-s + 0.625·23-s + 0.800·25-s + (0.278 + 0.482i)29-s + (0.449 + 0.777i)31-s + (−0.164 − 0.284i)37-s + (−0.234 + 0.405i)41-s + (0.0762 + 0.132i)43-s + (0.656 − 1.13i)47-s + (−0.412 + 0.713i)53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.0788 - 0.996i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.0788 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821892363\)
\(L(\frac12)\) \(\approx\) \(1.821892363\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.727983819894462360005376025673, −7.59138198208800436072287462643, −6.75703296297873166917452565292, −6.26729085358700877029278918108, −5.43595382165398384515773190355, −4.89552545801240067668619757131, −3.95909531820594637158109832633, −2.74961780244005007125561255462, −2.22224054598588813593369546812, −1.22122199731996706812437831191, 0.45649960206435032937598750492, 1.86008327995803345363273322274, 2.47725415776011799234868648549, 3.29515807268657889379469506677, 4.61782232909975529374115898003, 5.08281092781559029470218025238, 5.94734211382088224877077564373, 6.40123330487083928930825491305, 7.32937667828391958558298336383, 7.989963590901599018360246155508

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