Properties

Label 2-8041-1.1-c1-0-408
Degree $2$
Conductor $8041$
Sign $-1$
Analytic cond. $64.2077$
Root an. cond. $8.01297$
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  − 2.75·2-s + 0.843·3-s + 5.60·4-s + 3.43·5-s − 2.32·6-s − 0.344·7-s − 9.95·8-s − 2.28·9-s − 9.47·10-s − 11-s + 4.72·12-s − 4.61·13-s + 0.951·14-s + 2.89·15-s + 16.2·16-s + 17-s + 6.31·18-s + 7.54·19-s + 19.2·20-s − 0.290·21-s + 2.75·22-s + 1.39·23-s − 8.38·24-s + 6.80·25-s + 12.7·26-s − 4.45·27-s − 1.93·28-s + ⋯
L(s)  = 1  − 1.95·2-s + 0.486·3-s + 2.80·4-s + 1.53·5-s − 0.949·6-s − 0.130·7-s − 3.51·8-s − 0.763·9-s − 2.99·10-s − 0.301·11-s + 1.36·12-s − 1.28·13-s + 0.254·14-s + 0.748·15-s + 4.05·16-s + 0.242·17-s + 1.48·18-s + 1.73·19-s + 4.30·20-s − 0.0634·21-s + 0.588·22-s + 0.289·23-s − 1.71·24-s + 1.36·25-s + 2.49·26-s − 0.858·27-s − 0.365·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8041\)    =    \(11 \cdot 17 \cdot 43\)
Sign: $-1$
Analytic conductor: \(64.2077\)
Root analytic conductor: \(8.01297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8041,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 + T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 2.75T + 2T^{2} \)
3 \( 1 - 0.843T + 3T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
7 \( 1 + 0.344T + 7T^{2} \)
13 \( 1 + 4.61T + 13T^{2} \)
19 \( 1 - 7.54T + 19T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 - 3.89T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + 6.76T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 2.44T + 89T^{2} \)
97 \( 1 - 6.64T + 97T^{2} \)
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

−7.62285685329289845130144222837, −7.16016913412874437591106710332, −6.24313973298152332147215541255, −5.72829694206480849676632803344, −5.05539078521464542476764487322, −3.08935454391427154114152109848, −2.83454866623433394437592971511, −2.00719080357993155154216705092, −1.28600674847326691290381304229, 0, 1.28600674847326691290381304229, 2.00719080357993155154216705092, 2.83454866623433394437592971511, 3.08935454391427154114152109848, 5.05539078521464542476764487322, 5.72829694206480849676632803344, 6.24313973298152332147215541255, 7.16016913412874437591106710332, 7.62285685329289845130144222837

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