Properties

Degree $2$
Conductor $8016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.54·5-s + 4.05·7-s + 9-s − 0.833·11-s + 2.46·13-s − 2.54·15-s − 1.83·17-s + 5.92·19-s − 4.05·21-s + 3.92·23-s + 1.46·25-s − 27-s − 7.78·29-s + 8.46·31-s + 0.833·33-s + 10.3·35-s + 3.96·37-s − 2.46·39-s − 11.0·41-s + 10.1·43-s + 2.54·45-s − 0.498·47-s + 9.40·49-s + 1.83·51-s + 2.45·53-s − 2.12·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·5-s + 1.53·7-s + 0.333·9-s − 0.251·11-s + 0.682·13-s − 0.656·15-s − 0.446·17-s + 1.35·19-s − 0.883·21-s + 0.817·23-s + 0.293·25-s − 0.192·27-s − 1.44·29-s + 1.51·31-s + 0.145·33-s + 1.74·35-s + 0.651·37-s − 0.394·39-s − 1.72·41-s + 1.54·43-s + 0.379·45-s − 0.0726·47-s + 1.34·49-s + 0.257·51-s + 0.337·53-s − 0.285·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.979957658\)
\(L(\frac12)\) \(\approx\) \(2.979957658\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 - 4.05T + 7T^{2} \)
11 \( 1 + 0.833T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 - 5.92T + 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 - 8.46T + 31T^{2} \)
37 \( 1 - 3.96T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 0.498T + 47T^{2} \)
53 \( 1 - 2.45T + 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 3.97T + 71T^{2} \)
73 \( 1 - 6.83T + 73T^{2} \)
79 \( 1 + 7.34T + 79T^{2} \)
83 \( 1 - 4.66T + 83T^{2} \)
89 \( 1 + 4.56T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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

−7.70516828315442648298041333833, −7.23006423643045523813891868186, −6.20050936548757416229848350864, −5.77372804747824417124486456956, −5.02726615503979235841896992792, −4.65230990734082692927516236151, −3.53803433297097273134006423423, −2.44260823666896551179121071802, −1.64034711528165687739693352508, −0.972681135288972757009574280332, 0.972681135288972757009574280332, 1.64034711528165687739693352508, 2.44260823666896551179121071802, 3.53803433297097273134006423423, 4.65230990734082692927516236151, 5.02726615503979235841896992792, 5.77372804747824417124486456956, 6.20050936548757416229848350864, 7.23006423643045523813891868186, 7.70516828315442648298041333833

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