Properties

Label 2-8034-1.1-c1-0-8
Degree $2$
Conductor $8034$
Sign $1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3.67·5-s − 6-s − 3.11·7-s − 8-s + 9-s + 3.67·10-s + 2.60·11-s + 12-s + 13-s + 3.11·14-s − 3.67·15-s + 16-s − 3.87·17-s − 18-s − 8.68·19-s − 3.67·20-s − 3.11·21-s − 2.60·22-s + 1.84·23-s − 24-s + 8.50·25-s − 26-s + 27-s − 3.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.64·5-s − 0.408·6-s − 1.17·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s + 0.786·11-s + 0.288·12-s + 0.277·13-s + 0.832·14-s − 0.948·15-s + 0.250·16-s − 0.938·17-s − 0.235·18-s − 1.99·19-s − 0.821·20-s − 0.679·21-s − 0.556·22-s + 0.383·23-s − 0.204·24-s + 1.70·25-s − 0.196·26-s + 0.192·27-s − 0.588·28-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: $1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4187600890\)
\(L(\frac12)\) \(\approx\) \(0.4187600890\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.67T + 5T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 + 8.68T + 19T^{2} \)
23 \( 1 - 1.84T + 23T^{2} \)
29 \( 1 - 2.03T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 - 0.985T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + 0.0401T + 47T^{2} \)
53 \( 1 + 3.86T + 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + 9.47T + 61T^{2} \)
67 \( 1 + 5.87T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 - 2.66T + 73T^{2} \)
79 \( 1 + 0.518T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 - 1.83T + 89T^{2} \)
97 \( 1 + 2.45T + 97T^{2} \)
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

−7.87812723304935628252624475479, −7.27614604288166979102198581415, −6.53261117316157913272565824925, −6.29801050626113133422394195675, −4.72983905189524137939004504213, −4.02939535669339915624503366770, −3.52283735986832368666103700179, −2.77039432448796744722966932295, −1.70457262574009312378176944755, −0.33553933816055386407542027941, 0.33553933816055386407542027941, 1.70457262574009312378176944755, 2.77039432448796744722966932295, 3.52283735986832368666103700179, 4.02939535669339915624503366770, 4.72983905189524137939004504213, 6.29801050626113133422394195675, 6.53261117316157913272565824925, 7.27614604288166979102198581415, 7.87812723304935628252624475479

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