Properties

Label 2-460e2-1.1-c1-0-17
Degree $2$
Conductor $211600$
Sign $1$
Analytic cond. $1689.63$
Root an. cond. $41.1051$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·7-s + 6·9-s − 13-s − 6·21-s − 9·27-s − 3·29-s − 3·31-s − 8·37-s + 3·39-s + 3·41-s + 2·43-s − 11·47-s − 3·49-s − 14·53-s + 8·59-s + 4·61-s + 12·63-s + 4·67-s − 7·71-s + 9·73-s + 9·81-s − 4·83-s + 9·87-s + 2·89-s − 2·91-s + 9·93-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.755·7-s + 2·9-s − 0.277·13-s − 1.30·21-s − 1.73·27-s − 0.557·29-s − 0.538·31-s − 1.31·37-s + 0.480·39-s + 0.468·41-s + 0.304·43-s − 1.60·47-s − 3/7·49-s − 1.92·53-s + 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.488·67-s − 0.830·71-s + 1.05·73-s + 81-s − 0.439·83-s + 0.964·87-s + 0.211·89-s − 0.209·91-s + 0.933·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(211600\)    =    \(2^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1689.63\)
Root analytic conductor: \(41.1051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 211600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8999782204\)
\(L(\frac12)\) \(\approx\) \(0.8999782204\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−12.82732481424720, −12.57063843305134, −11.87656361637473, −11.57989818059086, −11.21560546156270, −10.82076245491319, −10.36973084548055, −9.804236161247565, −9.490913378415842, −8.686206991602432, −8.276320762643828, −7.503608891232912, −7.316718943552641, −6.652479663103066, −6.204574367447130, −5.747642158823935, −5.192893778009595, −4.784839244884510, −4.535699799554157, −3.652064656866985, −3.237065305591014, −2.040637519694272, −1.808367746583560, −0.9999828745508161, −0.3455532817004170, 0.3455532817004170, 0.9999828745508161, 1.808367746583560, 2.040637519694272, 3.237065305591014, 3.652064656866985, 4.535699799554157, 4.784839244884510, 5.192893778009595, 5.747642158823935, 6.204574367447130, 6.652479663103066, 7.316718943552641, 7.503608891232912, 8.276320762643828, 8.686206991602432, 9.490913378415842, 9.804236161247565, 10.36973084548055, 10.82076245491319, 11.21560546156270, 11.57989818059086, 11.87656361637473, 12.57063843305134, 12.82732481424720

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