↑

Properties

Label 1-709-709.140-r0-0-0
Degree $1$
Conductor $709$
Sign $0.907 - 0.420i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (−0.981 − 0.194i)2-s + (−0.560 + 0.828i)3-s + (0.924 + 0.380i)4-s + (0.999 + 0.0354i)5-s + (0.710 − 0.703i)6-s + (−0.746 − 0.665i)7-s + (−0.833 − 0.552i)8-s + (−0.372 − 0.928i)9-s + (−0.973 − 0.228i)10-s + (0.421 − 0.906i)11-s + (−0.833 + 0.552i)12-s + (0.949 + 0.314i)13-s + (0.603 + 0.797i)14-s + (−0.589 + 0.807i)15-s + (0.710 + 0.703i)16-s + (−0.996 − 0.0886i)17-s + ⋯
L(s)  = 1  + (−0.981 − 0.194i)2-s + (−0.560 + 0.828i)3-s + (0.924 + 0.380i)4-s + (0.999 + 0.0354i)5-s + (0.710 − 0.703i)6-s + (−0.746 − 0.665i)7-s + (−0.833 − 0.552i)8-s + (−0.372 − 0.928i)9-s + (−0.973 − 0.228i)10-s + (0.421 − 0.906i)11-s + (−0.833 + 0.552i)12-s + (0.949 + 0.314i)13-s + (0.603 + 0.797i)14-s + (−0.589 + 0.807i)15-s + (0.710 + 0.703i)16-s + (−0.996 − 0.0886i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.907 - 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7588444217 - 0.1674897411i\)
\(L(\frac12)\) \(\approx\) \(0.7588444217 - 0.1674897411i\)
\(L(1)\) \(\approx\) \(0.6699594487 + 0.001535624329i\)
\(L(1)\) \(\approx\) \(0.6699594487 + 0.001535624329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.981 - 0.194i)T \)
3 \( 1 + (-0.560 + 0.828i)T \)
5 \( 1 + (0.999 + 0.0354i)T \)
7 \( 1 + (-0.746 - 0.665i)T \)
11 \( 1 + (0.421 - 0.906i)T \)
13 \( 1 + (0.949 + 0.314i)T \)
17 \( 1 + (-0.996 - 0.0886i)T \)
19 \( 1 + (-0.0620 + 0.998i)T \)
23 \( 1 + (-0.722 + 0.691i)T \)
29 \( 1 + (0.758 + 0.651i)T \)
31 \( 1 + (-0.405 - 0.914i)T \)
37 \( 1 + (-0.746 - 0.665i)T \)
41 \( 1 + (0.781 - 0.624i)T \)
43 \( 1 + (0.515 - 0.857i)T \)
47 \( 1 + (-0.237 - 0.971i)T \)
53 \( 1 + (0.994 + 0.106i)T \)
59 \( 1 + (-0.833 - 0.552i)T \)
61 \( 1 + (0.984 - 0.176i)T \)
67 \( 1 + (0.631 - 0.775i)T \)
71 \( 1 + (-0.973 + 0.228i)T \)
73 \( 1 + (-0.887 + 0.461i)T \)
79 \( 1 + (0.545 - 0.838i)T \)
83 \( 1 + (0.802 + 0.596i)T \)
89 \( 1 + (0.00887 - 0.999i)T \)
97 \( 1 + (-0.0974 + 0.995i)T \)
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

−22.6449110143875383034769260458, −21.98679941696556961566277065594, −20.90562411922480048415195017428, −19.88649772721781495941788861120, −19.32800633047082776158958467490, −18.18998070099672326540056106592, −17.9267386345065364092356754858, −17.29299714472187652599875318729, −16.260075265356056296980337731290, −15.62123504432629091177083639306, −14.4757100732115178140813722936, −13.37389296280105258945760208748, −12.66912095023760555347524139473, −11.79756186039390858281294442265, −10.81854788535378095597847569559, −10.04296722079647065500570879815, −9.074153566231812235153383306994, −8.45851320258147004559436102983, −7.13603604920104983780902099130, −6.3701189955546515685053664016, −6.05117604656628605394016146860, −4.82814001378426863989289166767, −2.748929792869923203861937125519, −2.0798564323304070895829965017, −1.03241981716214175738231850807, 0.66206475921839568058118614528, 1.91210665259436315464472814831, 3.347682200918840678039679682141, 3.98017705277449255077293926893, 5.74318915100359794119543104392, 6.20258698409469620367296481739, 7.06040541040685185040476785200, 8.62935498242099609281115650184, 9.16997175075985384064858771028, 10.03016614281111032051954888782, 10.64519718825216866759480153833, 11.303148569936751396964444790497, 12.36994727401965954682748000107, 13.480655070863964175685103590023, 14.301915370973013178558763572541, 15.6630381886982132387569042010, 16.252678261405876207780253158782, 16.83627262231005837403374056472, 17.57623345938757968388225349997, 18.30992440088603455791655966040, 19.25791616740621218348134795795, 20.2448008376569608247469020459, 20.85423601725358389558386044445, 21.69575547565152350312959179222, 22.22141658837610821348407684261

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