Properties

Label 2-22848-1.1-c1-0-43
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 − 3·5-s + 7-s + 9-s + 11-s − 13-s + 3·15-s − 17-s + 6·19-s − 21-s + 2·23-s + 4·25-s − 27-s + 2·29-s − 33-s − 3·35-s − 5·37-s + 39-s + 4·41-s − 9·43-s − 3·45-s + 49-s + 51-s − 11·53-s − 3·55-s − 6·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.417·23-s + 4/5·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s − 0.507·35-s − 0.821·37-s + 0.160·39-s + 0.624·41-s − 1.37·43-s − 0.447·45-s + 1/7·49-s + 0.140·51-s − 1.51·53-s − 0.404·55-s − 0.794·57-s + 0.520·59-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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

−15.60981613941765, −15.45130075832274, −14.76209349314582, −14.20758232848206, −13.60285876341698, −12.96457627358595, −12.19488031716986, −11.94362431453269, −11.53787877650998, −10.90446940910913, −10.51910697977352, −9.608930914063482, −9.211499601922779, −8.330834776557479, −7.916968657437016, −7.329842444905165, −6.839678661816177, −6.165098898053451, −5.231997407026314, −4.887468730696072, −4.189691837216708, −3.514277575748094, −2.926803019119002, −1.747222139528291, −0.9146719091179843, 0, 0.9146719091179843, 1.747222139528291, 2.926803019119002, 3.514277575748094, 4.189691837216708, 4.887468730696072, 5.231997407026314, 6.165098898053451, 6.839678661816177, 7.329842444905165, 7.916968657437016, 8.330834776557479, 9.211499601922779, 9.608930914063482, 10.51910697977352, 10.90446940910913, 11.53787877650998, 11.94362431453269, 12.19488031716986, 12.96457627358595, 13.60285876341698, 14.20758232848206, 14.76209349314582, 15.45130075832274, 15.60981613941765

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