↑

Properties

Label 1-731-731.214-r0-0-0
Degree $1$
Conductor $731$
Sign $0.806 - 0.591i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (0.382 − 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s − i·13-s + (−0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s − 16-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (0.382 − 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s − i·13-s + (−0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8441976493 - 0.2765744864i\)
\(L(\frac12)\) \(\approx\) \(0.8441976493 - 0.2765744864i\)
\(L(1)\) \(\approx\) \(0.9017195231 + 0.1170987119i\)
\(L(1)\) \(\approx\) \(0.9017195231 + 0.1170987119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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

−22.34061613227737805070055983203, −21.8775231907165336760099937960, −21.12878651670412153694453446102, −20.32499181860488343365812875775, −19.37582497206309846385471784860, −19.07613757659890841469222453239, −17.80150780084753718320944355892, −16.41103909865315667632115536201, −15.95346579482303309613397720511, −15.44596390468972625589552568601, −14.36184548199803719573823553955, −13.47216376195287285666517833860, −12.47820462317507898322106630592, −11.71659311521568702374956268343, −11.192611141729748867297764439791, −10.26827022406528372624663175775, −9.22908865097859969554434181296, −8.785540213213135366401685824860, −7.06167494625079374641110217682, −6.04337205702344196263396473802, −5.16893197540697707387474292258, −4.39871115355113943380554117027, −3.38580229256863941054018131111, −2.935540098762968971047134182759, −0.99633208094939920752993093946, 0.43019899471257570657501523716, 2.45968217257779432034176938712, 3.278985318704672241821207491434, 4.33301267926321630870723132954, 5.42833997083717391517405519438, 6.34314407210318682364015006260, 7.163256029477948473053404590779, 7.64616150006091323079282155336, 8.46191769001053441342422245284, 10.01749922706010310324497654567, 11.03080381327196957276406435223, 12.07006502765105825474910803748, 12.626893170994709541999819733066, 13.19479947517855376354271095665, 14.257586052570643835722356754116, 15.04147611256956363978460571363, 15.96283222774168867078693839868, 16.51548466726299486734032569196, 17.58430296197039977360772877270, 18.21027616143605548786615396564, 19.1585977064511796975884469958, 20.03925074190463334030965764804, 20.70633030827595620393181997361, 22.33079392849447880845274298722, 22.67274726674421310608835381170

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