Properties

Label 2-85176-1.1-c1-0-3
Degree $2$
Conductor $85176$
Sign $1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 3·11-s − 4·17-s − 2·19-s − 23-s − 5·25-s − 4·29-s − 9·31-s − 3·37-s − 5·41-s + 4·43-s + 9·47-s + 49-s + 4·53-s + 10·59-s + 5·61-s − 11·67-s − 16·71-s − 11·73-s − 3·77-s − 5·79-s − 4·83-s − 2·89-s + 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.904·11-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 25-s − 0.742·29-s − 1.61·31-s − 0.493·37-s − 0.780·41-s + 0.609·43-s + 1.31·47-s + 1/7·49-s + 0.549·53-s + 1.30·59-s + 0.640·61-s − 1.34·67-s − 1.89·71-s − 1.28·73-s − 0.341·77-s − 0.562·79-s − 0.439·83-s − 0.211·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9006350432\)
\(L(\frac12)\) \(\approx\) \(0.9006350432\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.88026171013399, −13.38032897279120, −12.99922941232443, −12.50288365546342, −11.83426057025855, −11.53523254199470, −11.03184621022807, −10.28216813587110, −10.11263680838916, −9.210840073658888, −8.950705461243084, −8.633749844146254, −7.721766624523182, −7.224576584882995, −6.888339220099542, −6.111617281318475, −5.804767948945644, −5.177964753176024, −4.248747188306679, −4.027839262768933, −3.446649593533199, −2.595017558535467, −1.968892620875861, −1.411545434655068, −0.2968421810427553, 0.2968421810427553, 1.411545434655068, 1.968892620875861, 2.595017558535467, 3.446649593533199, 4.027839262768933, 4.248747188306679, 5.177964753176024, 5.804767948945644, 6.111617281318475, 6.888339220099542, 7.224576584882995, 7.721766624523182, 8.633749844146254, 8.950705461243084, 9.210840073658888, 10.11263680838916, 10.28216813587110, 11.03184621022807, 11.53523254199470, 11.83426057025855, 12.50288365546342, 12.99922941232443, 13.38032897279120, 13.88026171013399

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