Properties

Label 2-319e2-1.1-c1-0-7
Degree $2$
Conductor $101761$
Sign $1$
Analytic cond. $812.565$
Root an. cond. $28.5055$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 2·5-s + 7-s − 3·9-s + 4·10-s + 3·13-s − 2·14-s − 4·16-s − 4·17-s + 6·18-s − 4·19-s − 4·20-s − 9·23-s − 25-s − 6·26-s + 2·28-s + 4·31-s + 8·32-s + 8·34-s − 2·35-s − 6·36-s + 2·37-s + 8·38-s + 2·41-s + 10·43-s + 6·45-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s + 0.377·7-s − 9-s + 1.26·10-s + 0.832·13-s − 0.534·14-s − 16-s − 0.970·17-s + 1.41·18-s − 0.917·19-s − 0.894·20-s − 1.87·23-s − 1/5·25-s − 1.17·26-s + 0.377·28-s + 0.718·31-s + 1.41·32-s + 1.37·34-s − 0.338·35-s − 36-s + 0.328·37-s + 1.29·38-s + 0.312·41-s + 1.52·43-s + 0.894·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(101761\)    =    \(11^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(812.565\)
Root analytic conductor: \(28.5055\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 101761,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−14.18099666364790, −13.72560478968127, −13.43527588614987, −12.50242718075862, −12.12808689405845, −11.47332841370528, −11.25449330118392, −10.68907178133379, −10.48597204957234, −9.583655064247311, −9.230576502273447, −8.680354763191616, −8.229988082425556, −7.926082358962913, −7.626171033886330, −6.743005213410970, −6.188048067132161, −5.944862736089038, −4.883320101437540, −4.300187291871207, −3.985856128571465, −3.111458008196876, −2.357463733711372, −1.860558674138605, −1.059126420480536, 0, 0, 1.059126420480536, 1.860558674138605, 2.357463733711372, 3.111458008196876, 3.985856128571465, 4.300187291871207, 4.883320101437540, 5.944862736089038, 6.188048067132161, 6.743005213410970, 7.626171033886330, 7.926082358962913, 8.229988082425556, 8.680354763191616, 9.230576502273447, 9.583655064247311, 10.48597204957234, 10.68907178133379, 11.25449330118392, 11.47332841370528, 12.12808689405845, 12.50242718075862, 13.43527588614987, 13.72560478968127, 14.18099666364790

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