↑

Properties

Label 2-117-13.12-c11-0-36
Degree $2$
Conductor $117$
Sign $0.401 - 0.915i$
Analytic cond. $89.8961$
Root an. cond. $9.48135$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 63.1i·2-s − 1.93e3·4-s + 8.53e3i·5-s − 3.44e4i·7-s + 6.97e3i·8-s − 5.39e5·10-s − 9.98e3i·11-s + (5.37e5 − 1.22e6i)13-s + 2.17e6·14-s − 4.40e6·16-s − 2.39e6·17-s − 2.16e6i·19-s − 1.65e7i·20-s + 6.30e5·22-s + 4.46e7·23-s + ⋯
L(s)  = 1  + 1.39i·2-s − 0.946·4-s + 1.22i·5-s − 0.774i·7-s + 0.0752i·8-s − 1.70·10-s − 0.0186i·11-s + (0.401 − 0.915i)13-s + 1.08·14-s − 1.05·16-s − 0.409·17-s − 0.201i·19-s − 1.15i·20-s + 0.0260·22-s + 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.401 - 0.915i$
Analytic conductor: \(89.8961\)
Root analytic conductor: \(9.48135\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :11/2),\ 0.401 - 0.915i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.882585125\)
\(L(\frac12)\) \(\approx\) \(1.882585125\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-5.37e5 + 1.22e6i)T \)
good2 \( 1 - 63.1iT - 2.04e3T^{2} \)
5 \( 1 - 8.53e3iT - 4.88e7T^{2} \)
7 \( 1 + 3.44e4iT - 1.97e9T^{2} \)
11 \( 1 + 9.98e3iT - 2.85e11T^{2} \)
17 \( 1 + 2.39e6T + 3.42e13T^{2} \)
19 \( 1 + 2.16e6iT - 1.16e14T^{2} \)
23 \( 1 - 4.46e7T + 9.52e14T^{2} \)
29 \( 1 + 4.30e7T + 1.22e16T^{2} \)
31 \( 1 + 2.22e8iT - 2.54e16T^{2} \)
37 \( 1 + 8.02e8iT - 1.77e17T^{2} \)
41 \( 1 - 2.38e8iT - 5.50e17T^{2} \)
43 \( 1 + 2.50e8T + 9.29e17T^{2} \)
47 \( 1 - 4.54e8iT - 2.47e18T^{2} \)
53 \( 1 - 3.42e9T + 9.26e18T^{2} \)
59 \( 1 + 8.20e9iT - 3.01e19T^{2} \)
61 \( 1 - 1.48e9T + 4.35e19T^{2} \)
67 \( 1 + 1.83e9iT - 1.22e20T^{2} \)
71 \( 1 + 2.10e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.39e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.33e10T + 7.47e20T^{2} \)
83 \( 1 - 1.09e10iT - 1.28e21T^{2} \)
89 \( 1 - 7.04e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.76e10iT - 7.15e21T^{2} \)
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

−11.19003418863503009597921061472, −10.66660922485821285355901101463, −9.230418303372023304456936886525, −7.903521925490802203623737816124, −7.17349302084397208034583268790, −6.40343846396422211463666152222, −5.29957262586669074032854161476, −3.82977710891918381452983242220, −2.50187986925682454503849653433, −0.50553458840398305344561717964, 0.953246846716437254848193601616, 1.72699200029537545557637554968, 2.95760681090249923703773496828, 4.27047867714301132594285377825, 5.22585835918412754313071513204, 6.79399146003080104965482438733, 8.740334884205056080800898280930, 9.012639519281615109154617613386, 10.26394307424696404968988869212, 11.44338494415018832196940686479

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