↑

Properties

Label 1-547-547.520-r0-0-0
Degree $1$
Conductor $547$
Sign $0.956 - 0.292i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + 11-s + (−0.900 + 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + 11-s + (−0.900 + 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130986087 - 0.1693163815i\)
\(L(\frac12)\) \(\approx\) \(1.130986087 - 0.1693163815i\)
\(L(1)\) \(\approx\) \(0.9579923081 - 0.4310904753i\)
\(L(1)\) \(\approx\) \(0.9579923081 - 0.4310904753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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

−23.27089269536891662815538949961, −22.588444883171504299382651521121, −22.20836636639496040242576011369, −20.84820551026760156431764004053, −20.31982238091052274618211574621, −19.61863917224663245986512547299, −17.94968543981218916247474780182, −17.15799811758029234056752552038, −16.424540925428496647421894609086, −15.845481950830299245145096898207, −15.12591776376403588505703843128, −14.23156846487607194340623189185, −13.31820829385241322901597184291, −12.334098657764863471730568902965, −11.36055932345264833762038444129, −10.68325785096880089211697155161, −9.12192695729592925418728110945, −8.70701477352723651621992512076, −7.38400131747766339384784653923, −6.680398139882494103157146542180, −5.39766602326937708223394185087, −4.56169157263268717952161011523, −3.82720165254847517137993549699, −3.171801547819528519082401787311, −0.55654116084853959851711995230, 1.29270237164419720311712356175, 2.26990500246551617158321671257, 3.37441088189056633626414232093, 4.26076919528194699031739978561, 5.713669780509954713988470899953, 6.34036718220872483991339039380, 7.307960706947181641059137798100, 8.62981065317265735336159619057, 9.36123522099269284599394894443, 10.915717746914429475393898648708, 11.608823441766494972101220751705, 11.97032336951707680501636397191, 12.930669128382497522622383209956, 13.79192759907759560210652161210, 14.71128469338360624536791489088, 15.41371139775087424363831923524, 16.54707791640187696871701508630, 17.85980727607856315197167601432, 18.69116764876702860731866298127, 19.15569390131856160359941398608, 19.7589162991526421223832425160, 20.78703659867017724198135175862, 21.97511235458286289689070594570, 22.53506408901914744429184721894, 23.13668177531835223886307483709

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