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 − 0.676·5-s − 6-s − 2.72·7-s + 8-s + 9-s − 0.676·10-s − 5.27·11-s − 12-s − 13-s − 2.72·14-s + 0.676·15-s + 16-s + 1.01·17-s + 18-s + 4.84·19-s − 0.676·20-s + 2.72·21-s − 5.27·22-s − 5.02·23-s − 24-s − 4.54·25-s − 26-s − 27-s − 2.72·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.302·5-s − 0.408·6-s − 1.02·7-s + 0.353·8-s + 0.333·9-s − 0.213·10-s − 1.58·11-s − 0.288·12-s − 0.277·13-s − 0.727·14-s + 0.174·15-s + 0.250·16-s + 0.246·17-s + 0.235·18-s + 1.11·19-s − 0.151·20-s + 0.593·21-s − 1.12·22-s − 1.04·23-s − 0.204·24-s − 0.908·25-s − 0.196·26-s − 0.192·27-s − 0.514·28-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

\(L(1)\) \(\approx\) \(1.242745015\)
\(L(\frac12)\) \(\approx\) \(1.242745015\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.676T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
17 \( 1 - 1.01T + 17T^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + 5.02T + 23T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + 3.04T + 31T^{2} \)
37 \( 1 + 1.00T + 37T^{2} \)
41 \( 1 - 9.67T + 41T^{2} \)
43 \( 1 + 8.83T + 43T^{2} \)
47 \( 1 + 8.64T + 47T^{2} \)
53 \( 1 + 2.36T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 2.97T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 - 1.34T + 73T^{2} \)
79 \( 1 + 5.10T + 79T^{2} \)
83 \( 1 + 0.744T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 2.57T + 97T^{2} \)
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

−7.908601030893296992915499339553, −6.96108397264266839197697469883, −6.37010191384511330822561537795, −5.61603157078551500559581655614, −5.16867966948722730030942017657, −4.37145818529446937236489372767, −3.45535061631975882171926686890, −2.91735356066255310818460787211, −1.94955754800537447183933485177, −0.48221614469536022337110271216, 0.48221614469536022337110271216, 1.94955754800537447183933485177, 2.91735356066255310818460787211, 3.45535061631975882171926686890, 4.37145818529446937236489372767, 5.16867966948722730030942017657, 5.61603157078551500559581655614, 6.37010191384511330822561537795, 6.96108397264266839197697469883, 7.908601030893296992915499339553

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