Properties

Label 2-816-1.1-c1-0-11
Degree $2$
Conductor $816$
Sign $-1$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 17-s − 4·19-s − 25-s − 27-s − 10·29-s − 8·31-s − 4·33-s − 2·37-s + 2·39-s + 10·41-s − 12·43-s − 2·45-s − 7·49-s − 51-s + 6·53-s − 8·55-s + 4·57-s − 12·59-s − 10·61-s + 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-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 − 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 1.82·43-s − 0.298·45-s − 49-s − 0.140·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-1$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -1)\)

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

−9.749296880762886418033197739706, −9.080746654144001493159861980940, −7.972624440390951709752181868622, −7.22142790948264379459326388401, −6.37869009161194820234692371426, −5.36244298721167160718725340278, −4.24541437163957543556604196065, −3.56656081546400251035965478912, −1.78155952274398559703453953653, 0, 1.78155952274398559703453953653, 3.56656081546400251035965478912, 4.24541437163957543556604196065, 5.36244298721167160718725340278, 6.37869009161194820234692371426, 7.22142790948264379459326388401, 7.972624440390951709752181868622, 9.080746654144001493159861980940, 9.749296880762886418033197739706

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