Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
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 + 1.40·5-s − 6-s − 0.998·7-s − 8-s + 9-s − 1.40·10-s + 4.83·11-s + 12-s + 13-s + 0.998·14-s + 1.40·15-s + 16-s − 2.95·17-s − 18-s + 1.88·19-s + 1.40·20-s − 0.998·21-s − 4.83·22-s − 2.84·23-s − 24-s − 3.01·25-s − 26-s + 27-s − 0.998·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.629·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.445·10-s + 1.45·11-s + 0.288·12-s + 0.277·13-s + 0.266·14-s + 0.363·15-s + 0.250·16-s − 0.716·17-s − 0.235·18-s + 0.433·19-s + 0.314·20-s − 0.217·21-s − 1.02·22-s − 0.593·23-s − 0.204·24-s − 0.603·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-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

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 1.40T + 5T^{2} \)
7 \( 1 + 0.998T + 7T^{2} \)
11 \( 1 - 4.83T + 11T^{2} \)
17 \( 1 + 2.95T + 17T^{2} \)
19 \( 1 - 1.88T + 19T^{2} \)
23 \( 1 + 2.84T + 23T^{2} \)
29 \( 1 + 5.72T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 4.75T + 37T^{2} \)
41 \( 1 + 4.77T + 41T^{2} \)
43 \( 1 - 9.75T + 43T^{2} \)
47 \( 1 + 2.46T + 47T^{2} \)
53 \( 1 + 8.38T + 53T^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 + 6.11T + 61T^{2} \)
67 \( 1 + 4.55T + 67T^{2} \)
71 \( 1 + 3.66T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 1.76T + 83T^{2} \)
89 \( 1 + 0.860T + 89T^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.41984117638591593661979251013, −6.99943496764360535406612419085, −6.11445188964615441673884550635, −5.76830828101499172889131550931, −4.50377471473726538466494835070, −3.71254303226647613011329268814, −3.07910291002251247772113978990, −1.82916658007768075344499662719, −1.60374082069831512361915448633, 0, 1.60374082069831512361915448633, 1.82916658007768075344499662719, 3.07910291002251247772113978990, 3.71254303226647613011329268814, 4.50377471473726538466494835070, 5.76830828101499172889131550931, 6.11445188964615441673884550635, 6.99943496764360535406612419085, 7.41984117638591593661979251013

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