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Properties

Label 2-5292-63.4-c1-0-38
Degree $2$
Conductor $5292$
Sign $-0.603 + 0.797i$
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  + 2.46·5-s − 4.64·11-s + (3.55 − 6.15i)13-s + (2.25 − 3.90i)17-s + (2.16 + 3.74i)19-s − 5.86·23-s + 1.05·25-s + (−3.48 − 6.04i)29-s + (−3.69 − 6.39i)31-s + (0.363 + 0.629i)37-s + (−0.136 + 0.236i)41-s + (2.41 + 4.18i)43-s + (−1.83 + 3.18i)47-s + (2.52 − 4.37i)53-s − 11.4·55-s + ⋯
L(s)  = 1  + 1.10·5-s − 1.40·11-s + (0.985 − 1.70i)13-s + (0.547 − 0.948i)17-s + (0.496 + 0.859i)19-s − 1.22·23-s + 0.210·25-s + (−0.647 − 1.12i)29-s + (−0.662 − 1.14i)31-s + (0.0597 + 0.103i)37-s + (−0.0213 + 0.0369i)41-s + (0.368 + 0.638i)43-s + (−0.267 + 0.463i)47-s + (0.347 − 0.601i)53-s − 1.54·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.603 + 0.797i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.390439277\)
\(L(\frac12)\) \(\approx\) \(1.390439277\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.46T + 5T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (-3.55 + 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.25 + 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + (3.48 + 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.363 - 0.629i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.83 - 3.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.52 + 4.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.56 + 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.90 - 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + (2.16 - 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.21 - 5.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.742 + 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.91 + 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.246 + 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−7.74868339401918686657042057804, −7.61124654390258237285479201768, −6.14071001365310002344005845331, −5.66120775243088172286087039696, −5.46429120683860163669760684017, −4.23660752550577135301630869615, −3.20016548536248793317243310764, −2.56925407289908699513765379581, −1.57692145360440942996235076363, −0.32978324267864078608329804626, 1.51812174457366227306541200564, 2.01735471014825826009651439847, 3.10353095209949829608453373354, 3.98300734085128047715091857170, 4.93112692024474142556916885699, 5.65853107557561736615478421658, 6.15523946703284582606720653011, 6.98387121139295393181523998586, 7.67820698572682246311225814600, 8.610045337084878148648420891800

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