Properties

Label 2-8043-1.1-c1-0-145
Degree $2$
Conductor $8043$
Sign $1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.19·2-s + 3-s + 2.81·4-s + 2.36·5-s − 2.19·6-s − 7-s − 1.77·8-s + 9-s − 5.18·10-s − 0.0237·11-s + 2.81·12-s + 3.91·13-s + 2.19·14-s + 2.36·15-s − 1.72·16-s + 6.89·17-s − 2.19·18-s − 0.0907·19-s + 6.64·20-s − 21-s + 0.0520·22-s − 5.81·23-s − 1.77·24-s + 0.587·25-s − 8.59·26-s + 27-s − 2.81·28-s + ⋯
L(s)  = 1  − 1.55·2-s + 0.577·3-s + 1.40·4-s + 1.05·5-s − 0.895·6-s − 0.377·7-s − 0.628·8-s + 0.333·9-s − 1.63·10-s − 0.00714·11-s + 0.811·12-s + 1.08·13-s + 0.586·14-s + 0.610·15-s − 0.430·16-s + 1.67·17-s − 0.516·18-s − 0.0208·19-s + 1.48·20-s − 0.218·21-s + 0.0110·22-s − 1.21·23-s − 0.363·24-s + 0.117·25-s − 1.68·26-s + 0.192·27-s − 0.531·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627059147\)
\(L(\frac12)\) \(\approx\) \(1.627059147\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
383 \( 1 + T \)
good2 \( 1 + 2.19T + 2T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
11 \( 1 + 0.0237T + 11T^{2} \)
13 \( 1 - 3.91T + 13T^{2} \)
17 \( 1 - 6.89T + 17T^{2} \)
19 \( 1 + 0.0907T + 19T^{2} \)
23 \( 1 + 5.81T + 23T^{2} \)
29 \( 1 - 1.96T + 29T^{2} \)
31 \( 1 - 5.08T + 31T^{2} \)
37 \( 1 - 4.75T + 37T^{2} \)
41 \( 1 + 9.09T + 41T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 - 3.24T + 53T^{2} \)
59 \( 1 - 6.57T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 + 0.500T + 73T^{2} \)
79 \( 1 - 7.83T + 79T^{2} \)
83 \( 1 - 2.96T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 2.97T + 97T^{2} \)
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

−8.013627015028343051917558758601, −7.47036150721633726068745767986, −6.48842230594621539805869593917, −6.11684211057695060233204054091, −5.26750271634272180304305766120, −4.07366351467476591070113071172, −3.21898633354102738950588936690, −2.33672803480727654203164418533, −1.57973586496494061030439990073, −0.832009209370395025200408362837, 0.832009209370395025200408362837, 1.57973586496494061030439990073, 2.33672803480727654203164418533, 3.21898633354102738950588936690, 4.07366351467476591070113071172, 5.26750271634272180304305766120, 6.11684211057695060233204054091, 6.48842230594621539805869593917, 7.47036150721633726068745767986, 8.013627015028343051917558758601

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