↑

Properties

Label 1-712-712.403-r1-0-0
Degree $1$
Conductor $712$
Sign $0.989 + 0.144i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (−0.755 − 0.654i)3-s + (−0.959 + 0.281i)5-s + (0.281 + 0.959i)7-s + (0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.755 − 0.654i)13-s + (0.909 + 0.415i)15-s + (−0.415 − 0.909i)17-s + (−0.989 + 0.142i)19-s + (0.415 − 0.909i)21-s + (−0.989 + 0.142i)23-s + (0.841 − 0.540i)25-s + (0.540 − 0.841i)27-s + (−0.281 − 0.959i)29-s + (−0.989 − 0.142i)31-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (−0.959 + 0.281i)5-s + (0.281 + 0.959i)7-s + (0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.755 − 0.654i)13-s + (0.909 + 0.415i)15-s + (−0.415 − 0.909i)17-s + (−0.989 + 0.142i)19-s + (0.415 − 0.909i)21-s + (−0.989 + 0.142i)23-s + (0.841 − 0.540i)25-s + (0.540 − 0.841i)27-s + (−0.281 − 0.959i)29-s + (−0.989 − 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3130603208 + 0.02273342751i\)
\(L(\frac12)\) \(\approx\) \(0.3130603208 + 0.02273342751i\)
\(L(1)\) \(\approx\) \(0.4981710963 - 0.05890811129i\)
\(L(1)\) \(\approx\) \(0.4981710963 - 0.05890811129i\)

Euler product

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

−22.431749482561004488059237906288, −21.54534029288891231617358664406, −20.829211237781270059125275784466, −19.96944369167750624838468135928, −19.33411350737927043212030102963, −18.06669692505587351694021075170, −17.41763282618799870138589167364, −16.411540546998371396902266724657, −16.1454733104893079235440302581, −14.96720368756506549079340587540, −14.487835465065948135674561019923, −12.903510113051591198894198602413, −12.48077897580389226004612967961, −11.25995504665031746184358373871, −10.83751416388063708300113804618, −9.99378471646695271632675734342, −8.905738291934627852818405612369, −7.833219041756939269788374537295, −7.09502214983746848159140332881, −6.010004010941587091617606016834, −4.66207377594171641385636089955, −4.43361716246797473176406221227, −3.40681163984994236562641498463, −1.75098458352321336394530376396, −0.22997315289505465415263735625, 0.33796685822964025340857520531, 2.089037356293587166832058217888, 2.82552147666989458314401161665, 4.35158655315105621576361861765, 5.278454990947074297193024963488, 6.023851651900775119403875529009, 7.20120280259412556974260380942, 7.854624561015183705214386644805, 8.594303993059681954978686059490, 10.04250799644582794842050323114, 10.96240638585998702361639868031, 11.64096038477749595631772112978, 12.35676058397132740240357348144, 13.02333482877683933337927674438, 14.16600494168979287954214296408, 15.295079791454977482593019548177, 15.72476898137139509682828849706, 16.71115558853943093545534724900, 17.71916379459077739462129137580, 18.43704139628807171985223394564, 18.92434434360670556566751001310, 19.79351661428229456428613828620, 20.78675348092768074722435841458, 21.9520944288699352585783444826, 22.39121072115533886336162687868

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