Properties

Label 2-341e2-1.1-c1-0-0
Degree $2$
Conductor $116281$
Sign $1$
Analytic cond. $928.508$
Root an. cond. $30.4714$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 3·5-s − 2·9-s − 2·12-s − 3·15-s + 4·16-s + 6·20-s + 9·23-s + 4·25-s − 5·27-s + 4·36-s − 7·37-s + 6·45-s − 12·47-s + 4·48-s − 7·49-s − 6·53-s − 15·59-s + 6·60-s − 8·64-s + 13·67-s + 9·69-s − 3·71-s + 4·75-s − 12·80-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s − 0.577·12-s − 0.774·15-s + 16-s + 1.34·20-s + 1.87·23-s + 4/5·25-s − 0.962·27-s + 2/3·36-s − 1.15·37-s + 0.894·45-s − 1.75·47-s + 0.577·48-s − 49-s − 0.824·53-s − 1.95·59-s + 0.774·60-s − 64-s + 1.58·67-s + 1.08·69-s − 0.356·71-s + 0.461·75-s − 1.34·80-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116281\)    =    \(11^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(928.508\)
Root analytic conductor: \(30.4714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 116281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5201313905\)
\(L(\frac12)\) \(\approx\) \(0.5201313905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.59645932004375, −13.13429864980739, −12.67917597085912, −12.23202388278913, −11.66120726307688, −11.08855209796937, −10.93010360653516, −10.04709076216472, −9.563753995861224, −9.029050308509490, −8.616590357824690, −8.309331070682500, −7.676073091091474, −7.450132641791327, −6.625804770001535, −6.102447449911347, −5.184756444580701, −4.923195353645209, −4.426225847326836, −3.570139007951366, −3.367313608065065, −2.949656891849389, −1.915322139150954, −1.080384389334288, −0.2459114833353112, 0.2459114833353112, 1.080384389334288, 1.915322139150954, 2.949656891849389, 3.367313608065065, 3.570139007951366, 4.426225847326836, 4.923195353645209, 5.184756444580701, 6.102447449911347, 6.625804770001535, 7.450132641791327, 7.676073091091474, 8.309331070682500, 8.616590357824690, 9.029050308509490, 9.563753995861224, 10.04709076216472, 10.93010360653516, 11.08855209796937, 11.66120726307688, 12.23202388278913, 12.67917597085912, 13.13429864980739, 13.59645932004375

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