↑

Properties

Label 1-117-117.7-r1-0-0
Degree $1$
Conductor $117$
Sign $-0.782 + 0.622i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s − i·7-s − i·8-s + (0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·20-s + (−0.5 − 0.866i)22-s − 23-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s − i·7-s − i·8-s + (0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·20-s + (−0.5 − 0.866i)22-s − 23-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05705800789 - 0.1632867735i\)
\(L(\frac12)\) \(\approx\) \(-0.05705800789 - 0.1632867735i\)
\(L(1)\) \(\approx\) \(0.4775269840 - 0.2078560471i\)
\(L(1)\) \(\approx\) \(0.4775269840 - 0.2078560471i\)

Euler product

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

−29.46740357188836692258675739964, −27.99361973400194806270346457339, −27.71021218912184534918382176500, −26.54538209383664632465843024506, −25.634452374978888776167602030502, −24.621634047201789167381979611884, −23.71734429591855830714311613910, −22.59229556349639077272753701729, −21.39021764089984067619095458243, −19.83281549728586484881112677150, −19.09698828839539723580634617404, −18.37763864706039118705263513215, −17.0720403502135196767482377464, −16.03487796041069823334555099936, −15.063513475088110772548587830906, −14.34840429859795637848074909845, −12.26839328739881711974828921739, −11.35991842621585054264769937517, −10.19892496110001991353249036441, −8.79759850210370157014203142774, −8.05009554642082955605901652840, −6.6717610864917227821915097440, −5.68104317688147425643233485155, −3.72324739116367065222677340458, −1.94325417950066635788922195068, 0.09789623748655427703410216588, 1.479626575066653069236028428123, 3.46475336029042411519634635759, 4.48571983243372397666226622709, 6.83893264555325891953215921056, 7.706337964950546590411311689955, 8.886396772061897649177030287170, 10.00710489449219801028309508294, 11.23251447022904496841459830721, 12.09245990768742315929639012429, 13.22233751436928352876632931260, 14.82533786587257059722649760838, 16.25618528356439471469825848703, 16.8419662088538744191299621091, 17.98354449659425469787526645732, 19.2795213625761492521124771021, 20.0500645310082940076975115872, 20.64863790834474633305268359168, 22.11128222856413166903500131720, 23.28113287519697205341630447037, 24.35969076876881489858084722304, 25.556217851562621699092891119546, 26.53438628397763193978201874623, 27.62529759447227017204128815403, 27.91507088094968973241622648464

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