↑

Properties

Label 1-927-927.632-r1-0-0
Degree $1$
Conductor $927$
Sign $0.854 - 0.519i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.696 − 0.717i)2-s + (−0.0307 − 0.999i)4-s + (−0.332 + 0.943i)5-s + (−0.952 − 0.303i)7-s + (−0.739 − 0.673i)8-s + (0.445 + 0.895i)10-s + (0.696 − 0.717i)11-s + (0.213 + 0.976i)13-s + (−0.881 + 0.473i)14-s + (−0.998 + 0.0615i)16-s + (−0.932 + 0.361i)17-s + (−0.602 − 0.798i)19-s + (0.952 + 0.303i)20-s + (−0.0307 − 0.999i)22-s + (0.696 + 0.717i)23-s + ⋯
L(s)  = 1  + (0.696 − 0.717i)2-s + (−0.0307 − 0.999i)4-s + (−0.332 + 0.943i)5-s + (−0.952 − 0.303i)7-s + (−0.739 − 0.673i)8-s + (0.445 + 0.895i)10-s + (0.696 − 0.717i)11-s + (0.213 + 0.976i)13-s + (−0.881 + 0.473i)14-s + (−0.998 + 0.0615i)16-s + (−0.932 + 0.361i)17-s + (−0.602 − 0.798i)19-s + (0.952 + 0.303i)20-s + (−0.0307 − 0.999i)22-s + (0.696 + 0.717i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.951775909 - 0.5469387042i\)
\(L(\frac12)\) \(\approx\) \(1.951775909 - 0.5469387042i\)
\(L(1)\) \(\approx\) \(1.162690096 - 0.3960937493i\)
\(L(1)\) \(\approx\) \(1.162690096 - 0.3960937493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.696 - 0.717i)T \)
5 \( 1 + (-0.332 + 0.943i)T \)
7 \( 1 + (-0.952 - 0.303i)T \)
11 \( 1 + (0.696 - 0.717i)T \)
13 \( 1 + (0.213 + 0.976i)T \)
17 \( 1 + (-0.932 + 0.361i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (0.696 + 0.717i)T \)
29 \( 1 + (-0.650 - 0.759i)T \)
31 \( 1 + (0.552 + 0.833i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (-0.650 + 0.759i)T \)
43 \( 1 + (0.816 - 0.577i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.602 + 0.798i)T \)
59 \( 1 + (0.952 - 0.303i)T \)
61 \( 1 + (-0.779 - 0.626i)T \)
67 \( 1 + (-0.952 + 0.303i)T \)
71 \( 1 + (0.982 + 0.183i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (0.650 + 0.759i)T \)
83 \( 1 + (0.952 + 0.303i)T \)
89 \( 1 + (0.850 + 0.526i)T \)
97 \( 1 + (-0.153 - 0.988i)T \)
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

−22.09046854048440685064601684810, −20.82920650877538892916854116153, −20.39580242338797997094277730753, −19.55533273916840647670746183609, −18.47525263028196486112964608818, −17.45348350543880858911861843705, −16.77581718018797176945754000990, −16.16247021298665494609229565751, −15.262155140076035562142894322808, −14.88625897679833969218958990644, −13.492729934237102545646495134194, −12.97890741711961457936550856865, −12.34332736541834334515383228672, −11.656027702148306763357985426610, −10.302205900261384871302609463049, −9.12577603626101887427169747604, −8.65820630855866906827672715599, −7.62230082858717233294221898202, −6.71306452217693385400885707562, −5.93593111443110107576768902787, −4.99602955494284600193372321513, −4.17375924376134038643850720517, −3.35581937644025884708447833800, −2.168616419961304415524070753405, −0.51749557827964708285702369871, 0.66390802498213750932951667717, 2.04992733425678181182374075024, 2.99654002133737490878986611560, 3.78951081713077271324844613174, 4.41650134387239058884507595112, 5.94203027369977410414064991591, 6.53450118334210465667773697246, 7.17356119513224487043284793768, 8.85545233000194690005401682310, 9.45320177765489862713015086403, 10.570382356818080877905230062569, 11.12786340281620949740968564432, 11.76977567811225800801219509446, 12.83397905296082179234527060317, 13.64232807822357121483070624596, 14.13886747959079794635525460244, 15.198709393570199748127021276808, 15.710736227246705027644925935918, 16.781328260209504128824551952530, 17.820702528190710866850026611377, 18.946807295419199757807853556786, 19.33124045114741359071287106551, 19.75122490452078514401695433565, 20.9763659233120942675998213420, 21.88302409549089229155361466487

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