Properties

Label 2-32340-1.1-c1-0-37
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 − 2·17-s − 6·23-s + 25-s + 27-s − 2·29-s + 6·31-s − 33-s + 2·37-s − 2·39-s − 8·43-s + 45-s + 4·47-s − 2·51-s + 6·53-s − 55-s − 6·59-s − 8·61-s − 2·65-s + 10·67-s − 6·69-s − 6·73-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 − 0.485·17-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.149·45-s + 0.583·47-s − 0.280·51-s + 0.824·53-s − 0.134·55-s − 0.781·59-s − 1.02·61-s − 0.248·65-s + 1.22·67-s − 0.722·69-s − 0.702·73-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.25691308519723, −14.76373655253210, −14.24691954180150, −13.59874419109654, −13.44100502226048, −12.75716297658612, −12.06925331231146, −11.80755741049410, −10.91415574430772, −10.39111146210905, −9.945266774539954, −9.393974556929945, −8.890217326872373, −8.207419776061373, −7.771800830218964, −7.190527613266056, −6.385400592401438, −6.080666376731679, −5.112889715628859, −4.728490626848668, −3.915716787603583, −3.311495399495022, −2.377999767401255, −2.148832113974811, −1.118808359753476, 0, 1.118808359753476, 2.148832113974811, 2.377999767401255, 3.311495399495022, 3.915716787603583, 4.728490626848668, 5.112889715628859, 6.080666376731679, 6.385400592401438, 7.190527613266056, 7.771800830218964, 8.207419776061373, 8.890217326872373, 9.393974556929945, 9.945266774539954, 10.39111146210905, 10.91415574430772, 11.80755741049410, 12.06925331231146, 12.75716297658612, 13.44100502226048, 13.59874419109654, 14.24691954180150, 14.76373655253210, 15.25691308519723

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