Properties

Label 2-5292-63.25-c1-0-32
Degree $2$
Conductor $5292$
Sign $-0.401 + 0.915i$
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  + (0.119 − 0.207i)5-s + (−2.56 − 4.43i)11-s + (2.44 + 4.23i)13-s + (1.85 − 3.20i)17-s + (−1.83 − 3.16i)19-s + (3.71 − 6.42i)23-s + (2.47 + 4.28i)25-s + (1.73 − 3.00i)29-s − 0.717·31-s + (−2.30 − 3.98i)37-s + (2.80 + 4.85i)41-s + (−6.24 + 10.8i)43-s − 4.33·47-s + (−0.471 + 0.816i)53-s − 1.22·55-s + ⋯
L(s)  = 1  + (0.0534 − 0.0926i)5-s + (−0.772 − 1.33i)11-s + (0.677 + 1.17i)13-s + (0.449 − 0.777i)17-s + (−0.419 − 0.727i)19-s + (0.773 − 1.34i)23-s + (0.494 + 0.856i)25-s + (0.321 − 0.557i)29-s − 0.128·31-s + (−0.378 − 0.655i)37-s + (0.437 + 0.757i)41-s + (−0.952 + 1.64i)43-s − 0.633·47-s + (−0.0647 + 0.112i)53-s − 0.165·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.401 + 0.915i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.321161441\)
\(L(\frac12)\) \(\approx\) \(1.321161441\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.119 + 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.56 + 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.85 + 3.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 + 3.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.71 + 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.73 + 3.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.717T + 31T^{2} \)
37 \( 1 + (2.30 + 3.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.24 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.33T + 47T^{2} \)
53 \( 1 + (0.471 - 0.816i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.57T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
67 \( 1 + 0.660T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + (-1.83 + 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.74 + 6.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.060421972622498903873145543897, −7.21595999525090462675414498828, −6.47229063519674184908321428194, −5.91075492858531954760121156373, −4.94207074279321787206623360668, −4.42855972226345088081663119672, −3.22754255786117337470528362751, −2.75384511877871989359657800876, −1.46720508731898208279483342576, −0.36402150701682297550294853308, 1.22775864711823558420369696823, 2.15763229757690037494647571238, 3.19236478086197735760503865903, 3.84595081333773435549302617949, 4.94142551053796616936228273513, 5.42595909411358617435961802332, 6.25040857155314782685627344241, 7.07757268615536608544062975123, 7.73271703214368777837438446452, 8.318893882530040048047997825459

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