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 − 0.689·5-s − 6-s + 4.91·7-s − 8-s + 9-s + 0.689·10-s − 11-s + 12-s + 2.15·13-s − 4.91·14-s − 0.689·15-s + 16-s + 1.70·17-s − 18-s − 3.32·19-s − 0.689·20-s + 4.91·21-s + 22-s − 0.135·23-s − 24-s − 4.52·25-s − 2.15·26-s + 27-s + 4.91·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.308·5-s − 0.408·6-s + 1.85·7-s − 0.353·8-s + 0.333·9-s + 0.217·10-s − 0.301·11-s + 0.288·12-s + 0.598·13-s − 1.31·14-s − 0.177·15-s + 0.250·16-s + 0.413·17-s − 0.235·18-s − 0.763·19-s − 0.154·20-s + 1.07·21-s + 0.213·22-s − 0.0281·23-s − 0.204·24-s − 0.905·25-s − 0.423·26-s + 0.192·27-s + 0.929·28-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  =  \((2,\ 4026,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.148252336\)
\(L(\frac12)\)  \(\approx\)  \(2.148252336\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.689T + 5T^{2} \)
7 \( 1 - 4.91T + 7T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 3.32T + 19T^{2} \)
23 \( 1 + 0.135T + 23T^{2} \)
29 \( 1 - 4.83T + 29T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 - 3.02T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 1.60T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 9.93T + 71T^{2} \)
73 \( 1 + 1.42T + 73T^{2} \)
79 \( 1 - 0.110T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + 6.01T + 97T^{2} \)
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

−8.316148968916905419841884417988, −7.952085509993175132437548044683, −7.38259557837950351453049552885, −6.36847407320418527604176559731, −5.46991788360302041707688419486, −4.56014651097961851850364056339, −3.89404157900984236342158455809, −2.66986207861478079118925533436, −1.87147480329203994909869001847, −0.966005458342928826657615941027, 0.966005458342928826657615941027, 1.87147480329203994909869001847, 2.66986207861478079118925533436, 3.89404157900984236342158455809, 4.56014651097961851850364056339, 5.46991788360302041707688419486, 6.36847407320418527604176559731, 7.38259557837950351453049552885, 7.952085509993175132437548044683, 8.316148968916905419841884417988

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