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Properties

Label 2-32340-1.1-c1-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 11-s − 2·13-s + 15-s + 7·17-s − 3·19-s + 3·23-s + 25-s + 27-s − 5·29-s − 33-s + 2·37-s − 2·39-s + 43-s + 45-s − 8·47-s + 7·51-s − 9·53-s − 55-s − 3·57-s + 9·59-s − 5·61-s − 2·65-s − 2·67-s + 3·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s + 1.69·17-s − 0.688·19-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s − 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.152·43-s + 0.149·45-s − 1.16·47-s + 0.980·51-s − 1.23·53-s − 0.134·55-s − 0.397·57-s + 1.17·59-s − 0.640·61-s − 0.248·65-s − 0.244·67-s + 0.361·69-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: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−15.10438979993063, −14.67103114950882, −14.41456042152214, −13.79121356482229, −13.08029798170665, −12.84226413591445, −12.27673919485613, −11.60477481801181, −11.00805112635275, −10.37232902446684, −9.814635838927999, −9.572672652656523, −8.786963543356996, −8.302328394155429, −7.626410354159829, −7.277524649040036, −6.526141606803864, −5.809047076161342, −5.328143768906021, −4.645848928208219, −3.949228275754625, −3.122518184054258, −2.757174675378497, −1.829587406628243, −1.229807138080429, 0, 1.229807138080429, 1.829587406628243, 2.757174675378497, 3.122518184054258, 3.949228275754625, 4.645848928208219, 5.328143768906021, 5.809047076161342, 6.526141606803864, 7.277524649040036, 7.626410354159829, 8.302328394155429, 8.786963543356996, 9.572672652656523, 9.814635838927999, 10.37232902446684, 11.00805112635275, 11.60477481801181, 12.27673919485613, 12.84226413591445, 13.08029798170665, 13.79121356482229, 14.41456042152214, 14.67103114950882, 15.10438979993063

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