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Properties

Label 2-5292-63.58-c1-0-0
Degree $2$
Conductor $5292$
Sign $-0.963 + 0.266i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.963 + 0.266i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.963 + 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1622990662\)
\(L(\frac12)\) \(\approx\) \(0.1622990662\)
\(L(\frac{3}{2})\) 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

−8.745557306466319585242470256959, −7.67639724466243981539182566557, −6.99352563954319853062327988831, −6.85622032183614365256719068944, −5.49101870198102702910342188404, −4.79492745239463431312566288752, −4.47385760111187941793914158773, −3.17679210052322044946105783517, −2.41048228313046925994644427401, −1.51628835634084554840869990851, 0.04409564765249422588542056230, 1.16461036875003957458652084660, 2.59663354383671825979537453545, 3.08384311835473557406475616349, 4.00972488063878283382021516272, 5.05392179220907889468972730443, 5.57921104587497039427118465779, 6.29886098536433446877964380207, 7.13784193158360969633019683848, 7.969382773741590966878142808180

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