Properties

Label 2-32340-1.1-c1-0-11
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 + 5·13-s − 15-s − 2·17-s + 6·19-s + 2·23-s + 25-s + 27-s − 4·29-s − 9·31-s − 33-s + 6·37-s + 5·39-s + 12·41-s + 43-s − 45-s − 2·47-s − 2·51-s + 6·53-s + 55-s + 6·57-s − 59-s − 6·61-s − 5·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.38·13-s − 0.258·15-s − 0.485·17-s + 1.37·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.61·31-s − 0.174·33-s + 0.986·37-s + 0.800·39-s + 1.87·41-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 0.280·51-s + 0.824·53-s + 0.134·55-s + 0.794·57-s − 0.130·59-s − 0.768·61-s − 0.620·65-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: $\chi_{32340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.980087312\)
\(L(\frac12)\) \(\approx\) \(2.980087312\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−15.04091063003865, −14.53656482188095, −13.97125676098903, −13.49786676752600, −12.87397766806605, −12.69733407706246, −11.73650561905987, −11.22307757233672, −10.95325004824008, −10.26810330943048, −9.414879702908747, −9.160799087990298, −8.614396015832775, −7.848628374646434, −7.545817005180426, −6.975322392939906, −6.127795342213388, −5.646052927619779, −4.923000096282791, −4.118115757220442, −3.658088663741816, −3.067337160787781, −2.305476807509801, −1.427953701100894, −0.6686580364337093, 0.6686580364337093, 1.427953701100894, 2.305476807509801, 3.067337160787781, 3.658088663741816, 4.118115757220442, 4.923000096282791, 5.646052927619779, 6.127795342213388, 6.975322392939906, 7.545817005180426, 7.848628374646434, 8.614396015832775, 9.160799087990298, 9.414879702908747, 10.26810330943048, 10.95325004824008, 11.22307757233672, 11.73650561905987, 12.69733407706246, 12.87397766806605, 13.49786676752600, 13.97125676098903, 14.53656482188095, 15.04091063003865

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