Properties

Label 2-22848-1.1-c1-0-3
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 − 4·11-s − 2·13-s − 2·15-s − 17-s − 4·19-s + 21-s − 4·23-s − 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s − 2·35-s + 2·37-s + 2·39-s − 2·41-s + 8·43-s + 2·45-s + 49-s + 51-s − 6·53-s − 8·55-s + 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-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: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7228901117\)
\(L(\frac12)\) \(\approx\) \(0.7228901117\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.72213712948886, −14.86774293719580, −14.47167081215001, −13.74297006953496, −13.20474287741485, −12.86887581177115, −12.33125847650627, −11.72076480997971, −10.95019431006692, −10.48131517935180, −10.13965534337706, −9.468388065540831, −9.008650113696951, −8.146966127766118, −7.589338031300360, −6.977591029456121, −6.214838477501124, −5.831506065119092, −5.281918484682724, −4.587706048857350, −3.925196500366415, −2.909025379471115, −2.268041306966300, −1.662933360127116, −0.3340817701342571, 0.3340817701342571, 1.662933360127116, 2.268041306966300, 2.909025379471115, 3.925196500366415, 4.587706048857350, 5.281918484682724, 5.831506065119092, 6.214838477501124, 6.977591029456121, 7.589338031300360, 8.146966127766118, 9.008650113696951, 9.468388065540831, 10.13965534337706, 10.48131517935180, 10.95019431006692, 11.72076480997971, 12.33125847650627, 12.86887581177115, 13.20474287741485, 13.74297006953496, 14.47167081215001, 14.86774293719580, 15.72213712948886

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