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Properties

Label 2-5292-63.4-c1-0-28
Degree $2$
Conductor $5292$
Sign $-0.853 + 0.520i$
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  − 1.90·5-s + 3.06·11-s + (−1.13 + 1.96i)13-s + (−0.713 + 1.23i)17-s + (−2.98 − 5.16i)19-s + 7.15·23-s − 1.37·25-s + (−0.468 − 0.810i)29-s + (−4.11 − 7.11i)31-s + (−1.41 − 2.45i)37-s + (−5.31 + 9.20i)41-s + (2.98 + 5.16i)43-s + (0.483 − 0.837i)47-s + (−5.45 + 9.44i)53-s − 5.83·55-s + ⋯
L(s)  = 1  − 0.851·5-s + 0.924·11-s + (−0.313 + 0.543i)13-s + (−0.173 + 0.299i)17-s + (−0.684 − 1.18i)19-s + 1.49·23-s − 0.275·25-s + (−0.0869 − 0.150i)29-s + (−0.738 − 1.27i)31-s + (−0.232 − 0.403i)37-s + (−0.830 + 1.43i)41-s + (0.455 + 0.788i)43-s + (0.0705 − 0.122i)47-s + (−0.748 + 1.29i)53-s − 0.786·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.853 + 0.520i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.853 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3883816390\)
\(L(\frac12)\) \(\approx\) \(0.3883816390\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.90T + 5T^{2} \)
11 \( 1 - 3.06T + 11T^{2} \)
13 \( 1 + (1.13 - 1.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.713 - 1.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.98 + 5.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.15T + 23T^{2} \)
29 \( 1 + (0.468 + 0.810i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.11 + 7.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 + 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.98 - 5.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.483 + 0.837i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.45 - 9.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.68 - 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.449 + 0.778i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.813 + 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 + (-0.996 + 1.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.16 + 7.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.98 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.58 + 4.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.922 + 1.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−7.79087143959029284615338221639, −7.20399659697732063866958146152, −6.58344853140865841757266593769, −5.83246822855092615256159395633, −4.65565423671130571948450538913, −4.32233428374349547393879676625, −3.43400949987532089707866561739, −2.51549182337209865045781703857, −1.38615734502827953691378107968, −0.11216259812435011534925736041, 1.17962805535861681966616271100, 2.27763814619762114790451676008, 3.59832816757871221809422683383, 3.70860568535226447694560995675, 4.90877432724944263815140211288, 5.44496814980760012184625947871, 6.57173338468094690666932484907, 7.01053502096676242087619203741, 7.78028849479926087018163460391, 8.507107517023626313295274421408

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