Properties

Label 2-22848-1.1-c1-0-13
Degree $2$
Conductor $22848$
Sign $1$
Analytic cond. $182.442$
Root an. cond. $13.5071$
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 + 2·5-s − 7-s + 9-s − 2·13-s − 2·15-s + 17-s + 21-s − 25-s − 27-s − 2·29-s + 8·31-s − 2·35-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s − 51-s − 14·53-s + 8·59-s − 14·61-s − 63-s − 4·65-s − 4·67-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 0.242·17-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.140·51-s − 1.92·53-s + 1.04·59-s − 1.79·61-s − 0.125·63-s − 0.496·65-s − 0.488·67-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22848\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(182.442\)
Root analytic conductor: \(13.5071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695423534\)
\(L(\frac12)\) \(\approx\) \(1.695423534\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−15.41287677441730, −15.08691221226167, −14.31471800980936, −13.76326851662589, −13.39200989391989, −12.74408430206647, −12.25926515305847, −11.71414216708187, −11.13053651398333, −10.42714699512904, −9.991678123556882, −9.528368937447725, −9.066568239963030, −8.064019798523478, −7.718037153831619, −6.772205342512250, −6.360415876082822, −5.906193705154467, −5.086130981194733, −4.749555452530428, −3.801631534470349, −3.033281666116405, −2.259991339453603, −1.531441801154201, −0.5449310298138313, 0.5449310298138313, 1.531441801154201, 2.259991339453603, 3.033281666116405, 3.801631534470349, 4.749555452530428, 5.086130981194733, 5.906193705154467, 6.360415876082822, 6.772205342512250, 7.718037153831619, 8.064019798523478, 9.066568239963030, 9.528368937447725, 9.991678123556882, 10.42714699512904, 11.13053651398333, 11.71414216708187, 12.25926515305847, 12.74408430206647, 13.39200989391989, 13.76326851662589, 14.31471800980936, 15.08691221226167, 15.41287677441730

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