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 − 1.62·5-s − 6-s − 4.09·7-s − 8-s + 9-s + 1.62·10-s − 11-s + 12-s + 2.05·13-s + 4.09·14-s − 1.62·15-s + 16-s − 6.07·17-s − 18-s + 8.57·19-s − 1.62·20-s − 4.09·21-s + 22-s − 8.01·23-s − 24-s − 2.34·25-s − 2.05·26-s + 27-s − 4.09·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.728·5-s − 0.408·6-s − 1.54·7-s − 0.353·8-s + 0.333·9-s + 0.515·10-s − 0.301·11-s + 0.288·12-s + 0.569·13-s + 1.09·14-s − 0.420·15-s + 0.250·16-s − 1.47·17-s − 0.235·18-s + 1.96·19-s − 0.364·20-s − 0.893·21-s + 0.213·22-s − 1.67·23-s − 0.204·24-s − 0.469·25-s − 0.402·26-s + 0.192·27-s − 0.773·28-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  =  \((2,\ 4026,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.7906636501\)
\(L(\frac12)\)  \(\approx\)  \(0.7906636501\)
\(L(\frac{3}{2})\)   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 \( 1 + 1.62T + 5T^{2} \)
7 \( 1 + 4.09T + 7T^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 - 8.57T + 19T^{2} \)
23 \( 1 + 8.01T + 23T^{2} \)
29 \( 1 - 1.71T + 29T^{2} \)
31 \( 1 + 4.38T + 31T^{2} \)
37 \( 1 - 3.63T + 37T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 + 1.73T + 43T^{2} \)
47 \( 1 - 9.02T + 47T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 - 6.78T + 59T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 4.87T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 4.94T + 97T^{2} \)
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

−8.566128174546728644228659300948, −7.58679569304361691001922455155, −7.34676486904042190472210237317, −6.33022094834705844012647451997, −5.80657537115114780649784365064, −4.39745633283712581779086197603, −3.59479738287534794253280567729, −3.01103704302308926235174790304, −2.00642259728775169766992514351, −0.52437709068718499791329215528, 0.52437709068718499791329215528, 2.00642259728775169766992514351, 3.01103704302308926235174790304, 3.59479738287534794253280567729, 4.39745633283712581779086197603, 5.80657537115114780649784365064, 6.33022094834705844012647451997, 7.34676486904042190472210237317, 7.58679569304361691001922455155, 8.566128174546728644228659300948

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