Properties

Label 2-22848-1.1-c1-0-31
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 7-s + 9-s − 6·11-s + 2·15-s − 17-s − 2·19-s − 21-s − 25-s − 27-s + 4·29-s + 6·33-s − 2·35-s − 8·37-s + 2·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s + 51-s + 14·53-s + 12·55-s + 2·57-s − 6·59-s + 10·61-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.04·33-s − 0.338·35-s − 1.31·37-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 1.92·53-s + 1.61·55-s + 0.264·57-s − 0.781·59-s + 1.28·61-s + 0.125·63-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: \(1\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−15.74759435451892, −15.32136375334770, −14.98747734225921, −13.91242094333972, −13.72194667002657, −12.85348633093185, −12.50228900849046, −11.96724059417352, −11.31112213174999, −10.93343976606025, −10.31636411075803, −10.00125943156744, −8.995037109807529, −8.222518101997097, −8.132252828541716, −7.214817951072803, −6.962126433019163, −5.960056911740168, −5.373434942482353, −4.899395757448349, −4.207434025770579, −3.577732166508391, −2.640716481780136, −2.028332969360980, −0.7850663140986856, 0, 0.7850663140986856, 2.028332969360980, 2.640716481780136, 3.577732166508391, 4.207434025770579, 4.899395757448349, 5.373434942482353, 5.960056911740168, 6.962126433019163, 7.214817951072803, 8.132252828541716, 8.222518101997097, 8.995037109807529, 10.00125943156744, 10.31636411075803, 10.93343976606025, 11.31112213174999, 11.96724059417352, 12.50228900849046, 12.85348633093185, 13.72194667002657, 13.91242094333972, 14.98747734225921, 15.32136375334770, 15.74759435451892

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