Properties

Label 2-8034-1.1-c1-0-162
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 + 0.808·5-s + 6-s + 3.24·7-s + 8-s + 9-s + 0.808·10-s + 4.15·11-s + 12-s − 13-s + 3.24·14-s + 0.808·15-s + 16-s + 7.76·17-s + 18-s + 8.59·19-s + 0.808·20-s + 3.24·21-s + 4.15·22-s − 8.64·23-s + 24-s − 4.34·25-s − 26-s + 27-s + 3.24·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.361·5-s + 0.408·6-s + 1.22·7-s + 0.353·8-s + 0.333·9-s + 0.255·10-s + 1.25·11-s + 0.288·12-s − 0.277·13-s + 0.866·14-s + 0.208·15-s + 0.250·16-s + 1.88·17-s + 0.235·18-s + 1.97·19-s + 0.180·20-s + 0.707·21-s + 0.885·22-s − 1.80·23-s + 0.204·24-s − 0.869·25-s − 0.196·26-s + 0.192·27-s + 0.612·28-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

\(L(1)\) \(\approx\) \(6.328703921\)
\(L(\frac12)\) \(\approx\) \(6.328703921\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.808T + 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
17 \( 1 - 7.76T + 17T^{2} \)
19 \( 1 - 8.59T + 19T^{2} \)
23 \( 1 + 8.64T + 23T^{2} \)
29 \( 1 - 4.23T + 29T^{2} \)
31 \( 1 + 6.26T + 31T^{2} \)
37 \( 1 - 5.67T + 37T^{2} \)
41 \( 1 - 1.67T + 41T^{2} \)
43 \( 1 - 0.621T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 6.13T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 8.36T + 89T^{2} \)
97 \( 1 - 3.02T + 97T^{2} \)
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

−7.69629974924025119967501293541, −7.39735457171881171232249188195, −6.23825552856482932919542925142, −5.69221498404804033614938710409, −5.03381726765253322989866899377, −4.20836055450222414417837278437, −3.57735415395224714746562965598, −2.82101081508911525420748560590, −1.64076043929971618211285035487, −1.33799839497889227256628301565, 1.33799839497889227256628301565, 1.64076043929971618211285035487, 2.82101081508911525420748560590, 3.57735415395224714746562965598, 4.20836055450222414417837278437, 5.03381726765253322989866899377, 5.69221498404804033614938710409, 6.23825552856482932919542925142, 7.39735457171881171232249188195, 7.69629974924025119967501293541

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