Properties

Label 2-19320-1.1-c1-0-26
Degree $2$
Conductor $19320$
Sign $-1$
Analytic cond. $154.270$
Root an. cond. $12.4205$
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 + 7-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s − 23-s + 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 10·53-s − 4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(154.270\)
Root analytic conductor: \(12.4205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 19320,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.88582962243242, −15.28861433154526, −14.95185587735242, −14.33575448755656, −13.75250434876020, −13.38854999145867, −12.66546156213351, −12.37114149870741, −11.52038537280413, −10.93976434011239, −10.17748868855932, −10.03134311309943, −9.282486716067283, −8.545329000082698, −8.088428359183361, −7.616046270553266, −6.911683528648977, −6.168581531091734, −5.543287624117810, −4.801542132477957, −4.372110345104147, −3.355390404173569, −2.650532265030181, −2.161010215303590, −1.247710192829133, 0, 1.247710192829133, 2.161010215303590, 2.650532265030181, 3.355390404173569, 4.372110345104147, 4.801542132477957, 5.543287624117810, 6.168581531091734, 6.911683528648977, 7.616046270553266, 8.088428359183361, 8.545329000082698, 9.282486716067283, 10.03134311309943, 10.17748868855932, 10.93976434011239, 11.52038537280413, 12.37114149870741, 12.66546156213351, 13.38854999145867, 13.75250434876020, 14.33575448755656, 14.95185587735242, 15.28861433154526, 15.88582962243242

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