↑

Properties

Label 1-712-712.133-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.813 - 0.581i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (0.415 − 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (−0.142 + 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (0.415 − 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (−0.142 + 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.813 - 0.581i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.813 - 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5021537690 - 1.564837628i\)
\(L(\frac12)\) \(\approx\) \(0.5021537690 - 1.564837628i\)
\(L(1)\) \(\approx\) \(0.9852287994 - 0.7793833499i\)
\(L(1)\) \(\approx\) \(0.9852287994 - 0.7793833499i\)

Euler product

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

−22.78173426250901552188854741603, −21.8148638198867760998087409247, −21.411726673909498754592223833770, −20.88682503045756138092732884765, −19.39229470545359592711399842056, −18.98645181123939497806947537470, −18.29838959025382155326572554648, −16.97772969713727367760320507839, −16.345790479190366635122719161336, −15.40271579760326944852403081928, −14.63546418140051639627684551592, −14.247574130672612996407513532769, −13.18314881850680313993421654608, −11.81799950458120571619564623029, −11.18879857695745885768848022358, −10.347197135931779092519566591258, −9.525764193902426303113068676472, −8.57457896725826869728506769006, −7.98619590288533112598259009109, −6.376724210026780441108459342262, −5.915887290294177245008574986342, −4.66971415489480241535344482610, −3.57486925889256539292359062147, −2.88141988543642600505925967596, −1.85484628822770075318094932793, 0.78316989889683021782907170124, 1.5106595919241652482049719294, 2.789858835174942661288906315049, 3.963198094335832087775045332275, 4.97164231858495886632525843668, 5.98873871336178478598147379423, 7.228685867949952182491037591658, 7.651119550934223179006264851199, 8.69342671316483937676817784971, 9.47804111439936556436008374222, 10.49985454742017364610501433945, 11.65890410903984488223778051036, 12.53438433594753075837012808448, 13.17007497554093242866439860185, 13.75576237775875438734314769010, 14.77637904632420059420671853632, 15.635288416593458547494995930874, 16.89482942208737943287743629886, 17.34807547369795441955324951664, 18.07877641235604380442252867169, 19.16738786616465599902627735921, 20.03153490677193148587213254028, 20.43556486589726058673034124653, 21.07729509315639659528274988965, 22.50870832367718768609862421744

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