Properties

Label 2-22848-1.1-c1-0-41
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 − 2·11-s + 4·13-s + 2·15-s + 17-s + 2·19-s + 21-s + 4·23-s − 25-s − 27-s + 4·29-s − 8·31-s + 2·33-s + 2·35-s − 4·37-s − 4·39-s − 10·41-s + 8·43-s − 2·45-s + 49-s − 51-s + 6·53-s + 4·55-s − 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.516·15-s + 0.242·17-s + 0.458·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.348·33-s + 0.338·35-s − 0.657·37-s − 0.640·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s + 1/7·49-s − 0.140·51-s + 0.824·53-s + 0.539·55-s − 0.264·57-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.78791732535954, −15.30182843399200, −14.97245953186598, −13.86873281646534, −13.72978866317572, −12.82589312826602, −12.60302533583760, −11.81315219000664, −11.50002062564821, −10.79550797972431, −10.48669470907219, −9.802107595943182, −8.952572459482224, −8.640671194223991, −7.747869136373730, −7.415140136228511, −6.734482601995824, −6.076661510123060, −5.451146498955060, −4.896386932068311, −4.061520400023047, −3.514437262900001, −2.935885114679864, −1.790324649033484, −0.8884802685354980, 0, 0.8884802685354980, 1.790324649033484, 2.935885114679864, 3.514437262900001, 4.061520400023047, 4.896386932068311, 5.451146498955060, 6.076661510123060, 6.734482601995824, 7.415140136228511, 7.747869136373730, 8.640671194223991, 8.952572459482224, 9.802107595943182, 10.48669470907219, 10.79550797972431, 11.50002062564821, 11.81315219000664, 12.60302533583760, 12.82589312826602, 13.72978866317572, 13.86873281646534, 14.97245953186598, 15.30182843399200, 15.78791732535954

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