↑

Properties

Label 1-712-712.139-r1-0-0
Degree $1$
Conductor $712$
Sign $0.999 + 0.0228i$
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  + (0.142 − 0.989i)3-s + (−0.841 − 0.540i)5-s + (0.841 + 0.540i)7-s + (−0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.142 + 0.989i)13-s + (−0.654 + 0.755i)15-s + (−0.654 − 0.755i)17-s + (0.959 + 0.281i)19-s + (0.654 − 0.755i)21-s + (−0.959 − 0.281i)23-s + (0.415 + 0.909i)25-s + (−0.415 + 0.909i)27-s + (0.841 + 0.540i)29-s + (−0.959 + 0.281i)31-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (−0.841 − 0.540i)5-s + (0.841 + 0.540i)7-s + (−0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.142 + 0.989i)13-s + (−0.654 + 0.755i)15-s + (−0.654 − 0.755i)17-s + (0.959 + 0.281i)19-s + (0.654 − 0.755i)21-s + (−0.959 − 0.281i)23-s + (0.415 + 0.909i)25-s + (−0.415 + 0.909i)27-s + (0.841 + 0.540i)29-s + (−0.959 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.999 + 0.0228i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.999 + 0.0228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712598866 + 0.01955096806i\)
\(L(\frac12)\) \(\approx\) \(1.712598866 + 0.01955096806i\)
\(L(1)\) \(\approx\) \(1.018014264 - 0.2608490257i\)
\(L(1)\) \(\approx\) \(1.018014264 - 0.2608490257i\)

Euler product

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

−22.19901875157109876649427856700, −21.873824351856905765591643081125, −20.529034336625206841397653963482, −20.0479957112142746557013402188, −19.56170020781783740356759756756, −18.11346505093016186877312444607, −17.51201913848576970517855385088, −16.62541456684861705254391228568, −15.60648122696237604445049415115, −15.07073036655174307299073929202, −14.4243629381821616377323549801, −13.56040324279425621502909359813, −12.097342332744628462499120400, −11.46335970093021998817826127249, −10.63039935657323821656273492610, −10.01536649975542825416466880493, −8.85306281947066473347471997273, −7.97555255488243843239104805863, −7.274155981439451234336469331590, −5.991763942912953715908254863, −4.82905388417473608739219946717, −4.070621642956043840231966638813, −3.40331667544604798958344996416, −2.10409070424863639136635184098, −0.48382378794134800919223809035, 0.91202319846644979930572235479, 1.7365506136170364387068928118, 2.945659980652091481856682877727, 4.15820786184677166494799320682, 5.10408876802539999977414080526, 6.222746746629207126103051035935, 7.15233126450450782304592560855, 8.00877215048322577462494208154, 8.71325266299812225632149473009, 9.39579363184758202078873617563, 11.26301047106820673195073247064, 11.650561054398021109617780286051, 12.20354889161852283144434365846, 13.27258615361339185183137157347, 14.26146009856376200936073509764, 14.6650344372620508520658963818, 16.05898592896118957436481480769, 16.549155870584998972429385880904, 17.758107550707481769689247341904, 18.318964551787455957692933756293, 19.16841087851098079417744624495, 19.92899054091295670615155463467, 20.47222703664974380419658756010, 21.648125111188549823695169443458, 22.42373280610643807528135818193

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