Properties

Degree $2$
Conductor $8034$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2.74·5-s − 6-s + 3.39·7-s − 8-s + 9-s + 2.74·10-s − 0.860·11-s + 12-s + 13-s − 3.39·14-s − 2.74·15-s + 16-s − 4.14·17-s − 18-s − 2.70·19-s − 2.74·20-s + 3.39·21-s + 0.860·22-s + 0.937·23-s − 24-s + 2.54·25-s − 26-s + 27-s + 3.39·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.22·5-s − 0.408·6-s + 1.28·7-s − 0.353·8-s + 0.333·9-s + 0.868·10-s − 0.259·11-s + 0.288·12-s + 0.277·13-s − 0.906·14-s − 0.709·15-s + 0.250·16-s − 1.00·17-s − 0.235·18-s − 0.620·19-s − 0.614·20-s + 0.740·21-s + 0.183·22-s + 0.195·23-s − 0.204·24-s + 0.509·25-s − 0.196·26-s + 0.192·27-s + 0.641·28-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: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 + 2.74T + 5T^{2} \)
7 \( 1 - 3.39T + 7T^{2} \)
11 \( 1 + 0.860T + 11T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 - 0.937T + 23T^{2} \)
29 \( 1 - 0.521T + 29T^{2} \)
31 \( 1 - 7.56T + 31T^{2} \)
37 \( 1 + 0.373T + 37T^{2} \)
41 \( 1 + 5.16T + 41T^{2} \)
43 \( 1 + 9.15T + 43T^{2} \)
47 \( 1 - 3.73T + 47T^{2} \)
53 \( 1 - 6.82T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 4.99T + 61T^{2} \)
67 \( 1 - 7.00T + 67T^{2} \)
71 \( 1 + 9.02T + 71T^{2} \)
73 \( 1 + 8.63T + 73T^{2} \)
79 \( 1 - 1.21T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 4.60T + 89T^{2} \)
97 \( 1 + 8.14T + 97T^{2} \)
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

−7.78752874050129978697982478254, −7.01990844084742943404805911075, −6.43939640415112385431671173708, −5.23763253995086828613914723150, −4.50235697548725985033841230536, −3.95816094365698006633423483335, −2.97733057712756067823165408234, −2.13435392783651159413947587409, −1.24753528825188337223263338883, 0, 1.24753528825188337223263338883, 2.13435392783651159413947587409, 2.97733057712756067823165408234, 3.95816094365698006633423483335, 4.50235697548725985033841230536, 5.23763253995086828613914723150, 6.43939640415112385431671173708, 7.01990844084742943404805911075, 7.78752874050129978697982478254

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