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Properties

Label 2-32340-1.1-c1-0-8
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 + 6·17-s − 2·19-s + 25-s − 27-s + 4·31-s + 33-s − 10·37-s − 4·39-s − 4·43-s + 45-s − 12·47-s − 6·51-s + 6·53-s − 55-s + 2·57-s − 12·59-s + 10·61-s + 4·65-s − 4·67-s − 8·73-s − 75-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 + 1.45·17-s − 0.458·19-s + 1/5·25-s − 0.192·27-s + 0.718·31-s + 0.174·33-s − 1.64·37-s − 0.640·39-s − 0.609·43-s + 0.149·45-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.134·55-s + 0.264·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.936·73-s − 0.115·75-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

\(L(1)\) \(\approx\) \(2.092790116\)
\(L(\frac12)\) \(\approx\) \(2.092790116\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.13139254316499, −14.36363994513749, −14.08163990703432, −13.32143143177437, −13.00780446086716, −12.42334749743438, −11.72558069259162, −11.49511833389352, −10.62632902447303, −10.25082358370303, −9.944449093712292, −9.061885493129684, −8.563490555445134, −8.004457872879512, −7.332952764959181, −6.683327904517481, −6.088677197153973, −5.702342163891817, −5.013072675182015, −4.492869721726810, −3.452826184030231, −3.226188541296689, −2.029182378901136, −1.435971430308393, −0.5937232363659155, 0.5937232363659155, 1.435971430308393, 2.029182378901136, 3.226188541296689, 3.452826184030231, 4.492869721726810, 5.013072675182015, 5.702342163891817, 6.088677197153973, 6.683327904517481, 7.332952764959181, 8.004457872879512, 8.563490555445134, 9.061885493129684, 9.944449093712292, 10.25082358370303, 10.62632902447303, 11.49511833389352, 11.72558069259162, 12.42334749743438, 13.00780446086716, 13.32143143177437, 14.08163990703432, 14.36363994513749, 15.13139254316499

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