↑

Properties

Label 1-1441-1441.170-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.970 + 0.239i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.836 + 0.548i)2-s + (0.958 + 0.285i)3-s + (0.399 + 0.916i)4-s + (−0.943 + 0.331i)5-s + (0.644 + 0.764i)6-s + (−0.0724 + 0.997i)7-s + (−0.168 + 0.985i)8-s + (0.836 + 0.548i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.644 − 0.764i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (−0.681 + 0.732i)16-s + (−0.681 + 0.732i)17-s + (0.399 + 0.916i)18-s + ⋯
L(s)  = 1  + (0.836 + 0.548i)2-s + (0.958 + 0.285i)3-s + (0.399 + 0.916i)4-s + (−0.943 + 0.331i)5-s + (0.644 + 0.764i)6-s + (−0.0724 + 0.997i)7-s + (−0.168 + 0.985i)8-s + (0.836 + 0.548i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.644 − 0.764i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (−0.681 + 0.732i)16-s + (−0.681 + 0.732i)17-s + (0.399 + 0.916i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.970 + 0.239i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3377171618 + 2.781381352i\)
\(L(\frac12)\) \(\approx\) \(0.3377171618 + 2.781381352i\)
\(L(1)\) \(\approx\) \(1.353962047 + 1.334073793i\)
\(L(1)\) \(\approx\) \(1.353962047 + 1.334073793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.836 + 0.548i)T \)
3 \( 1 + (0.958 + 0.285i)T \)
5 \( 1 + (-0.943 + 0.331i)T \)
7 \( 1 + (-0.0724 + 0.997i)T \)
13 \( 1 + (0.644 - 0.764i)T \)
17 \( 1 + (-0.681 + 0.732i)T \)
19 \( 1 + (-0.906 + 0.421i)T \)
23 \( 1 + (0.885 - 0.464i)T \)
29 \( 1 + (0.779 + 0.626i)T \)
31 \( 1 + (-0.989 + 0.144i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (-0.906 + 0.421i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (0.779 - 0.626i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.906 - 0.421i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (-0.443 - 0.896i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.0241 + 0.999i)T \)
83 \( 1 + (0.215 + 0.976i)T \)
89 \( 1 + T \)
97 \( 1 + (0.644 - 0.764i)T \)
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

−20.26760466062420377152029883806, −19.84958413085641505002178001005, −19.1554347544085789636524807034, −18.62292182133051750243050649340, −17.39170892465118865554651697586, −16.19336472227107303424554748878, −15.72947802609736699182837915002, −14.87417197525712707049105949517, −14.18615584276618505282116066528, −13.38291135201910248647510276472, −13.01047753211595771988239322847, −12.07282622583454519284055892584, −11.18851330076375995928611693949, −10.66618883504191113852020267176, −9.39570054931224664697744521336, −8.89701793299499396184878059197, −7.706290106301812322328970375569, −7.050077895851531499666932065911, −6.35230228327156850107172831386, −4.77436694497133705512217825061, −4.2298585273089192105362592351, −3.60597511790655450913370394559, −2.7173708704562738093925132467, −1.635585935888954403054754355064, −0.67779251590138929134230650948, 1.87744234711938743927106349195, 2.889338435893353343518330617863, 3.40343755668601926615104425521, 4.27947267804389174508507496101, 5.05169737753033421077054872784, 6.19367253405854138226504097519, 6.91605846583125553753693762626, 7.96083394270051687306962715171, 8.4597002922619824329496367737, 9.00050719913635802158630696559, 10.55957817087967830064322988364, 11.054171994365627357511276720500, 12.31510146943911133038004230008, 12.68121885349228832845288807211, 13.54544569096910850657522012280, 14.52792724276606928668055009338, 15.17144042472192888074372716131, 15.38217040129929707711085088125, 16.14163680058212690792764667039, 17.036177592320815205563320831242, 18.284883583544896468737312655963, 18.82089278593543928294476630878, 19.79142206733948912363607411616, 20.35585502784798430555373455017, 21.18098589556859118041642369582

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