Properties

Label 2-8043-1.1-c1-0-219
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.723·2-s − 3-s − 1.47·4-s + 0.387·5-s − 0.723·6-s − 7-s − 2.51·8-s + 9-s + 0.280·10-s − 4.50·11-s + 1.47·12-s + 4.47·13-s − 0.723·14-s − 0.387·15-s + 1.13·16-s + 0.605·17-s + 0.723·18-s + 0.0501·19-s − 0.572·20-s + 21-s − 3.26·22-s + 1.96·23-s + 2.51·24-s − 4.84·25-s + 3.24·26-s − 27-s + 1.47·28-s + ⋯
L(s)  = 1  + 0.511·2-s − 0.577·3-s − 0.738·4-s + 0.173·5-s − 0.295·6-s − 0.377·7-s − 0.889·8-s + 0.333·9-s + 0.0887·10-s − 1.35·11-s + 0.426·12-s + 1.24·13-s − 0.193·14-s − 0.100·15-s + 0.282·16-s + 0.146·17-s + 0.170·18-s + 0.0115·19-s − 0.127·20-s + 0.218·21-s − 0.695·22-s + 0.409·23-s + 0.513·24-s − 0.969·25-s + 0.635·26-s − 0.192·27-s + 0.278·28-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: \(1\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 + T \)
383 \( 1 + T \)
good2 \( 1 - 0.723T + 2T^{2} \)
5 \( 1 - 0.387T + 5T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 0.605T + 17T^{2} \)
19 \( 1 - 0.0501T + 19T^{2} \)
23 \( 1 - 1.96T + 23T^{2} \)
29 \( 1 + 0.953T + 29T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 - 3.84T + 37T^{2} \)
41 \( 1 + 5.49T + 41T^{2} \)
43 \( 1 - 0.404T + 43T^{2} \)
47 \( 1 - 4.72T + 47T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 0.369T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
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

−7.54777384019298986449006983136, −6.41594904693164934813946414515, −6.05733551543151536296473363166, −5.30780007524444731738482557298, −4.84260325939600576520154046136, −3.92177332507948738957437681749, −3.32453466554922168487325655128, −2.38359666026242626331689882690, −1.04968116480653272205730700716, 0, 1.04968116480653272205730700716, 2.38359666026242626331689882690, 3.32453466554922168487325655128, 3.92177332507948738957437681749, 4.84260325939600576520154046136, 5.30780007524444731738482557298, 6.05733551543151536296473363166, 6.41594904693164934813946414515, 7.54777384019298986449006983136

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