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Properties

Label 2-32340-1.1-c1-0-13
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 − 13-s − 15-s + 4·17-s + 6·19-s − 23-s + 25-s + 27-s + 5·29-s − 33-s − 39-s + 3·41-s + 43-s − 45-s + 7·47-s + 4·51-s − 3·53-s + 55-s + 6·57-s + 8·59-s − 6·61-s + 65-s + 14·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.970·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 0.174·33-s − 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s + 1.02·47-s + 0.560·51-s − 0.412·53-s + 0.134·55-s + 0.794·57-s + 1.04·59-s − 0.768·61-s + 0.124·65-s + 1.71·67-s − 0.120·69-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

\(L(1)\) \(\approx\) \(2.951216515\)
\(L(\frac12)\) \(\approx\) \(2.951216515\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−15.10779334176847, −14.40666258523054, −13.96333221289356, −13.74392728240345, −12.77131265245708, −12.49471397013168, −11.92553530786894, −11.38369737583878, −10.77131867886772, −10.04350349927873, −9.767182652930284, −9.110175228131418, −8.485319176496777, −7.845830727814892, −7.605377691337136, −6.934984363087929, −6.273207488220274, −5.366898728070710, −5.096651325313425, −4.147659301109169, −3.656777464581133, −2.925002424720590, −2.452039106092458, −1.372349340726131, −0.6757608377176765, 0.6757608377176765, 1.372349340726131, 2.452039106092458, 2.925002424720590, 3.656777464581133, 4.147659301109169, 5.096651325313425, 5.366898728070710, 6.273207488220274, 6.934984363087929, 7.605377691337136, 7.845830727814892, 8.485319176496777, 9.110175228131418, 9.767182652930284, 10.04350349927873, 10.77131867886772, 11.38369737583878, 11.92553530786894, 12.49471397013168, 12.77131265245708, 13.74392728240345, 13.96333221289356, 14.40666258523054, 15.10779334176847

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