Properties

Label 2-32340-1.1-c1-0-18
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 11-s + 3·13-s − 15-s + 2·17-s + 2·23-s + 25-s − 27-s + 6·29-s + 11·31-s + 33-s + 4·37-s − 3·39-s − 2·41-s + 11·43-s + 45-s + 6·47-s − 2·51-s − 4·53-s − 55-s − 3·59-s + 8·61-s + 3·65-s + 8·67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.258·15-s + 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.97·31-s + 0.174·33-s + 0.657·37-s − 0.480·39-s − 0.312·41-s + 1.67·43-s + 0.149·45-s + 0.875·47-s − 0.280·51-s − 0.549·53-s − 0.134·55-s − 0.390·59-s + 1.02·61-s + 0.372·65-s + 0.977·67-s − 0.240·69-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: $\chi_{32340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.718246853\)
\(L(\frac12)\) \(\approx\) \(2.718246853\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.30771121806007, −14.30474668933189, −13.99722442748596, −13.52784815467244, −12.79652887302702, −12.51475794680956, −11.82899693129229, −11.29805854202794, −10.80263829280962, −10.20678813827291, −9.856484691472407, −9.150103632788531, −8.513398503930268, −8.012273921904018, −7.355201924469848, −6.584020638861820, −6.228903127017151, −5.651089690388246, −4.999507891883741, −4.438503016090979, −3.722053832303292, −2.865273749205553, −2.309697211692543, −1.173140892405636, −0.7702991286837434, 0.7702991286837434, 1.173140892405636, 2.309697211692543, 2.865273749205553, 3.722053832303292, 4.438503016090979, 4.999507891883741, 5.651089690388246, 6.228903127017151, 6.584020638861820, 7.355201924469848, 8.012273921904018, 8.513398503930268, 9.150103632788531, 9.856484691472407, 10.20678813827291, 10.80263829280962, 11.29805854202794, 11.82899693129229, 12.51475794680956, 12.79652887302702, 13.52784815467244, 13.99722442748596, 14.30474668933189, 15.30771121806007

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