Properties

Label 2-229320-1.1-c1-0-16
Degree $2$
Conductor $229320$
Sign $1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  + 5-s + 2·11-s + 13-s − 4·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 2·31-s + 6·37-s − 2·43-s − 8·47-s + 12·53-s + 2·55-s − 4·59-s + 2·61-s + 65-s − 16·67-s + 12·71-s − 6·73-s − 4·83-s − 4·85-s − 4·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.986·37-s − 0.304·43-s − 1.16·47-s + 1.64·53-s + 0.269·55-s − 0.520·59-s + 0.256·61-s + 0.124·65-s − 1.95·67-s + 1.42·71-s − 0.702·73-s − 0.439·83-s − 0.433·85-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.884747732\)
\(L(\frac12)\) \(\approx\) \(1.884747732\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−13.04592043472865, −12.53310508293875, −11.80354311121465, −11.73716475909589, −10.99075880018448, −10.65727379258804, −10.11019524696887, −9.696085919365570, −9.178686573044467, −8.661361992010538, −8.377143094330508, −7.780699911341620, −7.139019933753257, −6.638848908309720, −6.190953602786407, −5.954165743790058, −5.132526664757264, −4.654476712444640, −4.074851724010976, −3.771384706766247, −2.839935433386775, −2.454154643666637, −1.772801408296169, −1.270016984672218, −0.3758071111245694, 0.3758071111245694, 1.270016984672218, 1.772801408296169, 2.454154643666637, 2.839935433386775, 3.771384706766247, 4.074851724010976, 4.654476712444640, 5.132526664757264, 5.954165743790058, 6.190953602786407, 6.638848908309720, 7.139019933753257, 7.780699911341620, 8.377143094330508, 8.661361992010538, 9.178686573044467, 9.696085919365570, 10.11019524696887, 10.65727379258804, 10.99075880018448, 11.73716475909589, 11.80354311121465, 12.53310508293875, 13.04592043472865

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