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Properties

Label 2-32340-1.1-c1-0-14
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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  + 3-s + 5-s + 9-s + 11-s + 4·13-s + 15-s + 2·17-s − 2·19-s − 8·23-s + 25-s + 27-s − 4·29-s + 4·31-s + 33-s − 2·37-s + 4·39-s + 12·41-s + 8·43-s + 45-s + 4·47-s + 2·51-s − 10·53-s + 55-s − 2·57-s − 4·59-s − 6·61-s + 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s + 1.87·41-s + 1.21·43-s + 0.149·45-s + 0.583·47-s + 0.280·51-s − 1.37·53-s + 0.134·55-s − 0.264·57-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.664973639\)
\(L(\frac12)\) \(\approx\) \(3.664973639\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.09219448559995, −14.21884598800323, −14.03875561777932, −13.74656490693546, −12.79491549557895, −12.61438885536323, −11.98591396448958, −11.07930887076628, −10.92122444678182, −10.12032652319635, −9.593553049026740, −9.181790178577783, −8.547836002963537, −7.955614741488484, −7.606024841891754, −6.705901301758420, −6.092657073280834, −5.835109646672275, −4.918979159808012, −4.038715329718409, −3.831737812289306, −2.917153710261776, −2.210832831511986, −1.567395211760324, −0.7167267793461253, 0.7167267793461253, 1.567395211760324, 2.210832831511986, 2.917153710261776, 3.831737812289306, 4.038715329718409, 4.918979159808012, 5.835109646672275, 6.092657073280834, 6.705901301758420, 7.606024841891754, 7.955614741488484, 8.547836002963537, 9.181790178577783, 9.593553049026740, 10.12032652319635, 10.92122444678182, 11.07930887076628, 11.98591396448958, 12.61438885536323, 12.79491549557895, 13.74656490693546, 14.03875561777932, 14.21884598800323, 15.09219448559995

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