Properties

Degree $2$
Conductor $8034$
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  − 2-s − 3-s + 4-s + 1.59·5-s + 6-s + 4.24·7-s − 8-s + 9-s − 1.59·10-s + 4.07·11-s − 12-s + 13-s − 4.24·14-s − 1.59·15-s + 16-s − 4.06·17-s − 18-s + 3.43·19-s + 1.59·20-s − 4.24·21-s − 4.07·22-s + 9.26·23-s + 24-s − 2.44·25-s − 26-s − 27-s + 4.24·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.714·5-s + 0.408·6-s + 1.60·7-s − 0.353·8-s + 0.333·9-s − 0.505·10-s + 1.22·11-s − 0.288·12-s + 0.277·13-s − 1.13·14-s − 0.412·15-s + 0.250·16-s − 0.985·17-s − 0.235·18-s + 0.788·19-s + 0.357·20-s − 0.927·21-s − 0.867·22-s + 1.93·23-s + 0.204·24-s − 0.488·25-s − 0.196·26-s − 0.192·27-s + 0.802·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.122338280\)
\(L(\frac12)\) \(\approx\) \(2.122338280\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 4.07T + 11T^{2} \)
17 \( 1 + 4.06T + 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 - 9.26T + 23T^{2} \)
29 \( 1 - 7.09T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 5.89T + 37T^{2} \)
41 \( 1 + 5.28T + 41T^{2} \)
43 \( 1 + 1.53T + 43T^{2} \)
47 \( 1 + 9.61T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 - 6.49T + 61T^{2} \)
67 \( 1 - 4.52T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 3.94T + 73T^{2} \)
79 \( 1 - 1.53T + 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 + 8.97T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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

−7.85669764246791791031757915575, −7.02429011645471106880981139650, −6.66632694918691482109796560102, −5.74226096077915821415202316595, −5.13186721130199020727196099854, −4.49153402811783841181827560507, −3.50064411428505984103175736172, −2.25979395875445229529535715970, −1.53495714481227442292984935248, −0.931947859253951152177642075376, 0.931947859253951152177642075376, 1.53495714481227442292984935248, 2.25979395875445229529535715970, 3.50064411428505984103175736172, 4.49153402811783841181827560507, 5.13186721130199020727196099854, 5.74226096077915821415202316595, 6.66632694918691482109796560102, 7.02429011645471106880981139650, 7.85669764246791791031757915575

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