Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 61 $
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 + 2.48·5-s − 6-s + 1.57·7-s − 8-s + 9-s − 2.48·10-s − 11-s + 12-s − 4.97·13-s − 1.57·14-s + 2.48·15-s + 16-s + 7.22·17-s − 18-s + 5.46·19-s + 2.48·20-s + 1.57·21-s + 22-s − 1.75·23-s − 24-s + 1.16·25-s + 4.97·26-s + 27-s + 1.57·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.11·5-s − 0.408·6-s + 0.596·7-s − 0.353·8-s + 0.333·9-s − 0.784·10-s − 0.301·11-s + 0.288·12-s − 1.37·13-s − 0.421·14-s + 0.640·15-s + 0.250·16-s + 1.75·17-s − 0.235·18-s + 1.25·19-s + 0.555·20-s + 0.344·21-s + 0.213·22-s − 0.366·23-s − 0.204·24-s + 0.232·25-s + 0.975·26-s + 0.192·27-s + 0.298·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4026,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.375859662\)
\(L(\frac12)\)  \(\approx\)  \(2.375859662\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;61\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 + T \)
61 \( 1 + T \)
good5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 - 7.22T + 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 + 1.75T + 23T^{2} \)
29 \( 1 - 2.32T + 29T^{2} \)
31 \( 1 + 2.23T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 0.0841T + 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 - 0.738T + 53T^{2} \)
59 \( 1 - 6.61T + 59T^{2} \)
67 \( 1 - 7.08T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 0.359T + 83T^{2} \)
89 \( 1 - 4.63T + 89T^{2} \)
97 \( 1 - 5.94T + 97T^{2} \)
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

−8.349902830216676724111823939326, −7.74891798410272132053523653488, −7.33214473089070579099038545505, −6.30286045391391046598228483177, −5.40074361545595361970594996761, −4.97673425380646067391230394658, −3.56019868620478195573989182361, −2.66471693224979398723133324248, −1.95961061094092407107240920579, −1.00241819620386381406635628560, 1.00241819620386381406635628560, 1.95961061094092407107240920579, 2.66471693224979398723133324248, 3.56019868620478195573989182361, 4.97673425380646067391230394658, 5.40074361545595361970594996761, 6.30286045391391046598228483177, 7.33214473089070579099038545505, 7.74891798410272132053523653488, 8.349902830216676724111823939326

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