↑

Properties

Label 1-5896-5896.131-r0-0-0
Degree $1$
Conductor $5896$
Sign $0.702 - 0.711i$
Analytic cond. $27.3809$
Root an. cond. $27.3809$
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  + (−0.959 − 0.281i)3-s + (0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)13-s + (−0.841 + 0.540i)15-s + (−0.415 + 0.909i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)21-s + (0.959 + 0.281i)23-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + 29-s + (0.142 − 0.989i)31-s + (0.959 − 0.281i)35-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)3-s + (0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)13-s + (−0.841 + 0.540i)15-s + (−0.415 + 0.909i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)21-s + (0.959 + 0.281i)23-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + 29-s + (0.142 − 0.989i)31-s + (0.959 − 0.281i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5896\)    =    \(2^{3} \cdot 11 \cdot 67\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(27.3809\)
Root analytic conductor: \(27.3809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5896} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5896,\ (0:\ ),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451942945 - 0.6065214198i\)
\(L(\frac12)\) \(\approx\) \(1.451942945 - 0.6065214198i\)
\(L(1)\) \(\approx\) \(0.9734766582 - 0.1757619521i\)
\(L(1)\) \(\approx\) \(0.9734766582 - 0.1757619521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
67 \( 1 \)
good3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + T \)
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

−17.66167345174010163488685200936, −17.16823822508398139937156936345, −16.85817135583522475120760894035, −15.78723766383932802285967385495, −15.33995763079805289529530373645, −14.461076166683254429644005871681, −13.927917793908190712070688991839, −13.40014809100363635879655278710, −12.35394933164308432235473864214, −11.78112769098924223796970927548, −11.0378194289258932309190853189, −10.651082438582281269583119653868, −10.13335132307543359297803425451, −9.161831993274904938485047668847, −8.69808179384123172032033535502, −7.37659631173516687426529606338, −6.8375712187399277685298741924, −6.578102358777543668269158279327, −5.46704362649614088296615965932, −4.88654906947717677955922315878, −4.35853581663047693076031415533, −3.42319374334563089233354852588, −2.397958182152019463284047586840, −1.6873843195328719469894216646, −0.755163108916076858256939407657, 0.61074270810740475539690447004, 1.54161129948813371384464350192, 1.96479960931288350375084547838, 3.013721099098641682047619518815, 4.3410538880750109147456875805, 4.76792149474856941493097948109, 5.48695634791870113015350546785, 6.01025547661387606911235033035, 6.603117711874394272032969167980, 7.718280349718870871579195780398, 8.24737274237993962545532635412, 8.87121589888891428014598947538, 9.82449368964636893657858326712, 10.47155813748914559446774081650, 11.00311740868955752285158482050, 11.83299624475765860332083840483, 12.42143815240412653927825434017, 12.95390036695343835383045093218, 13.45597531895609412326771413941, 14.42033819272921834114422472556, 15.20510632131154577620649468175, 15.64079281449725520306796127050, 16.68863345969439465802868295745, 17.05357432620293780350470092471, 17.673588677093605849463485134772

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