↑

Properties

Label 1-927-927.263-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.803 + 0.595i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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  − 2-s + 4-s + 5-s + (−0.5 + 0.866i)7-s − 8-s − 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 20-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s + (−0.5 + 0.866i)7-s − 8-s − 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 20-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2000004957 + 0.6056150186i\)
\(L(\frac12)\) \(\approx\) \(0.2000004957 + 0.6056150186i\)
\(L(1)\) \(\approx\) \(0.6344347485 + 0.2269364097i\)
\(L(1)\) \(\approx\) \(0.6344347485 + 0.2269364097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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

−21.30548179360543400005520162835, −20.691288561975860415522199939372, −19.87483585051264450637781636172, −19.138376853687337406608745199166, −18.34071493288226969646359482021, −17.45119626331217991008221243305, −17.061519967109477821948102275586, −16.19602402977383540672571659081, −15.44850993344944148337196380541, −14.35134514844732299065910485910, −13.44801775527348828440120433000, −12.831620656962851301250656257486, −11.603078919129948007357691468922, −10.63178555471795272847715883737, −10.20775125363097506444133241762, −9.368479479470121169259420941540, −8.60122419397663060992811053951, −7.45467512848215247050666844969, −6.91005409444236326173576347413, −5.83355671890294016678408530537, −5.10812574717850535907786359767, −3.3045014153006654393762969017, −2.74080038999203423524194406826, −1.44132560599841631190510942807, −0.36518123265716042736460925834, 1.68599047432296142278092643900, 2.15930331810829961710598325615, 3.17906698196248286504263460879, 4.76248892859616484791807155640, 5.91942350217666516834939665175, 6.4036047391156139841164556518, 7.42838021727406043113391260415, 8.45612274188407001544517385133, 9.24057959359673021739055633990, 9.92682926112697770809594444136, 10.44812236181555910039925102661, 11.66086591323075788024382173309, 12.51267223843530222928754610255, 13.06796798508134761066394319202, 14.67320241417804776894932406872, 14.87277541737521597510346153924, 16.09425945639900529813217655233, 16.82314559775227755310076956559, 17.39038265813694975466709494296, 18.38808459870700702985152088180, 18.810975723402931530627952682911, 19.566849475972099390400757047321, 20.837003229098123067630446803126, 21.04363016244560939538992694955, 21.990686779046672747522922573854

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