Properties

Label 2-32340-1.1-c1-0-25
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 11-s − 5·13-s − 15-s + 2·17-s − 19-s + 2·23-s + 25-s − 27-s + 6·29-s − 31-s + 33-s + 37-s + 5·39-s + 8·41-s + 43-s + 45-s + 2·47-s − 2·51-s − 4·53-s − 55-s + 57-s + 6·59-s − 14·61-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 − 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.179·31-s + 0.174·33-s + 0.164·37-s + 0.800·39-s + 1.24·41-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 0.280·51-s − 0.549·53-s − 0.134·55-s + 0.132·57-s + 0.781·59-s − 1.79·61-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: \(1\)
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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.23324034895885, −14.80767608734396, −14.25319686540705, −13.77532313095351, −13.10813073520318, −12.48343299989571, −12.33238972817372, −11.59996925178836, −11.06496657094033, −10.29419241450827, −10.20760765797594, −9.402346974365497, −9.025642535678872, −8.157007172995611, −7.582324449634893, −7.100620612442994, −6.470658169381880, −5.786517326133449, −5.385244199672510, −4.592429910611511, −4.326454675159875, −3.100923749435616, −2.672810180952757, −1.819259102305770, −0.9590146692747762, 0, 0.9590146692747762, 1.819259102305770, 2.672810180952757, 3.100923749435616, 4.326454675159875, 4.592429910611511, 5.385244199672510, 5.786517326133449, 6.470658169381880, 7.100620612442994, 7.582324449634893, 8.157007172995611, 9.025642535678872, 9.402346974365497, 10.20760765797594, 10.29419241450827, 11.06496657094033, 11.59996925178836, 12.33238972817372, 12.48343299989571, 13.10813073520318, 13.77532313095351, 14.25319686540705, 14.80767608734396, 15.23324034895885

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