↑

Properties

Label 1-117-117.95-r1-0-0
Degree $1$
Conductor $117$
Sign $0.944 - 0.329i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ 0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.427371623 - 0.5817043652i\)
\(L(\frac12)\) \(\approx\) \(3.427371623 - 0.5817043652i\)
\(L(1)\) \(\approx\) \(2.085212009 - 0.1959125376i\)
\(L(1)\) \(\approx\) \(2.085212009 - 0.1959125376i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.59200583753848036615768641655, −27.98046338610810628304864024596, −26.91508728218746165532010031779, −25.891063708576006300243076290684, −24.7329551419102662465337362408, −23.669843474285949364417167205770, −22.93508557306845798239564382098, −22.05810345686618617914373623597, −20.93292169739913308925594984706, −19.879178313837448319128778471227, −18.999647416923577428188243609848, −17.36531848343602351666559461961, −16.36570649450617278492491331734, −14.93245191515044830870097867592, −14.452728320136641245533930895991, −13.337516040002076291869964392526, −11.89238751788743506225807230742, −11.14978369887325133860424527141, −10.052832478792924947489882472690, −7.88521381968738081726138694103, −7.03743384546344129880305677557, −5.81686422909288709047681234687, −4.149433464257809983583978894766, −3.43107393564383522955875171215, −1.57591627724811411582934508644, 1.34090902871091111756438641467, 3.02566093943063315005621156636, 4.52885830513698511424664035012, 5.329332234815875663516046692512, 6.790258393942993835953603786816, 8.17250276521063459621330051785, 9.391251114386093759911016633557, 11.34410839686814790360148792935, 11.925209487680657597038333098078, 12.92461219227665777473090590024, 14.20609858192312180274601027842, 15.19371426597113131394629437793, 16.14817694626408163685754499179, 17.15756264638828514720260094276, 18.794477091911090818354376545508, 19.99342278800781674352021054114, 20.75120461323300862884878453179, 21.8262787928231645655151995865, 22.73578034174747180538685007636, 23.91798002473662277628610515905, 24.67903301046175882984587411639, 25.31653829448427867874529718826, 27.054045901676044893309899306195, 28.1091669852299350520874876781, 28.86751324558816619622805732818

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