Properties

Label 2-5292-1.1-c1-0-21
Degree $2$
Conductor $5292$
Sign $1$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 5.29·11-s + 5.29·13-s + 3·17-s + 5.29·19-s − 4·25-s + 5.29·29-s − 5.29·31-s − 3·37-s + 9·41-s + 43-s − 47-s − 10.5·53-s − 5.29·55-s + 11·59-s − 5.29·61-s − 5.29·65-s + 4·67-s − 10.5·71-s + 10.5·73-s − 11·79-s + 9·83-s − 3·85-s − 14·89-s − 5.29·95-s − 5.29·97-s + 14·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.59·11-s + 1.46·13-s + 0.727·17-s + 1.21·19-s − 0.800·25-s + 0.982·29-s − 0.950·31-s − 0.493·37-s + 1.40·41-s + 0.152·43-s − 0.145·47-s − 1.45·53-s − 0.713·55-s + 1.43·59-s − 0.677·61-s − 0.656·65-s + 0.488·67-s − 1.25·71-s + 1.23·73-s − 1.23·79-s + 0.987·83-s − 0.325·85-s − 1.48·89-s − 0.542·95-s − 0.537·97-s + 1.39·101-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.371750916\)
\(L(\frac12)\) \(\approx\) \(2.371750916\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 5T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 - 11T + 59T^{2} \)
61 \( 1 + 5.29T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−8.170589397014392166543196132406, −7.52175203973653761773453979047, −6.73397743879514499597523237693, −6.06711559433425280646855244616, −5.41550773392264935561544538830, −4.27911240260640510592430490805, −3.72996169023300728089180923275, −3.10026115458149180359649435255, −1.62629689979229770365084772629, −0.924076880969759459726011204651, 0.924076880969759459726011204651, 1.62629689979229770365084772629, 3.10026115458149180359649435255, 3.72996169023300728089180923275, 4.27911240260640510592430490805, 5.41550773392264935561544538830, 6.06711559433425280646855244616, 6.73397743879514499597523237693, 7.52175203973653761773453979047, 8.170589397014392166543196132406

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