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Properties

Label 2-32340-1.1-c1-0-36
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 − 4·13-s − 15-s + 5·17-s + 5·19-s − 3·23-s + 25-s + 27-s − 29-s + 2·31-s + 33-s − 4·37-s − 4·39-s + 10·41-s − 9·43-s − 45-s − 6·47-s + 5·51-s − 7·53-s − 55-s + 5·57-s + 59-s − 5·61-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 + 1.21·17-s + 1.14·19-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.359·31-s + 0.174·33-s − 0.657·37-s − 0.640·39-s + 1.56·41-s − 1.37·43-s − 0.149·45-s − 0.875·47-s + 0.700·51-s − 0.961·53-s − 0.134·55-s + 0.662·57-s + 0.130·59-s − 0.640·61-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−15.23497787962640, −14.66209540521828, −14.31894039154385, −13.91720994277215, −13.21129194338963, −12.60332868332344, −12.10132267202698, −11.76265625320706, −11.14693051921570, −10.30512497121875, −9.878724601388165, −9.492530403687056, −8.844759690329740, −8.117709536503607, −7.689474426675840, −7.315582501235293, −6.618908628885276, −5.856594864396291, −5.182163667600899, −4.635626687834543, −3.898846790408614, −3.232294917243123, −2.806466342946083, −1.833354505220215, −1.110478617446216, 0, 1.110478617446216, 1.833354505220215, 2.806466342946083, 3.232294917243123, 3.898846790408614, 4.635626687834543, 5.182163667600899, 5.856594864396291, 6.618908628885276, 7.315582501235293, 7.689474426675840, 8.117709536503607, 8.844759690329740, 9.492530403687056, 9.878724601388165, 10.30512497121875, 11.14693051921570, 11.76265625320706, 12.10132267202698, 12.60332868332344, 13.21129194338963, 13.91720994277215, 14.31894039154385, 14.66209540521828, 15.23497787962640

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