↑

Properties

Label 1-927-927.290-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.989 - 0.143i$
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  + (−0.445 − 0.895i)2-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.332 − 0.943i)7-s + (0.982 + 0.183i)8-s + (0.273 − 0.961i)10-s + (−0.998 − 0.0615i)11-s + (0.332 − 0.943i)13-s + (−0.992 + 0.122i)14-s + (−0.273 − 0.961i)16-s + (−0.816 + 0.577i)17-s + (−0.0307 − 0.999i)19-s + (−0.982 + 0.183i)20-s + (0.389 + 0.920i)22-s + (−0.552 − 0.833i)23-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.332 − 0.943i)7-s + (0.982 + 0.183i)8-s + (0.273 − 0.961i)10-s + (−0.998 − 0.0615i)11-s + (0.332 − 0.943i)13-s + (−0.992 + 0.122i)14-s + (−0.273 − 0.961i)16-s + (−0.816 + 0.577i)17-s + (−0.0307 − 0.999i)19-s + (−0.982 + 0.183i)20-s + (0.389 + 0.920i)22-s + (−0.552 − 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.989 - 0.143i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.989 - 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04765108059 - 0.6602880376i\)
\(L(\frac12)\) \(\approx\) \(0.04765108059 - 0.6602880376i\)
\(L(1)\) \(\approx\) \(0.6535459166 - 0.3851784265i\)
\(L(1)\) \(\approx\) \(0.6535459166 - 0.3851784265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.445 - 0.895i)T \)
5 \( 1 + (0.739 + 0.673i)T \)
7 \( 1 + (0.332 - 0.943i)T \)
11 \( 1 + (-0.998 - 0.0615i)T \)
13 \( 1 + (0.332 - 0.943i)T \)
17 \( 1 + (-0.816 + 0.577i)T \)
19 \( 1 + (-0.0307 - 0.999i)T \)
23 \( 1 + (-0.552 - 0.833i)T \)
29 \( 1 + (-0.739 - 0.673i)T \)
31 \( 1 + (-0.969 - 0.243i)T \)
37 \( 1 + (-0.932 + 0.361i)T \)
41 \( 1 + (0.952 + 0.303i)T \)
43 \( 1 + (-0.932 - 0.361i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.881 + 0.473i)T \)
59 \( 1 + (-0.332 - 0.943i)T \)
61 \( 1 + (0.816 - 0.577i)T \)
67 \( 1 + (0.982 + 0.183i)T \)
71 \( 1 + (-0.952 - 0.303i)T \)
73 \( 1 + (-0.739 + 0.673i)T \)
79 \( 1 + (0.213 - 0.976i)T \)
83 \( 1 + (0.982 - 0.183i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (0.0922 - 0.995i)T \)
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

−22.226761241453008412650102860379, −21.45080445073024662471941715674, −20.76126907991035474386714603748, −19.77482339305664873418464348036, −18.70454030624894673976764427554, −18.132750441447953848859493147263, −17.661092936977900723668836701534, −16.407652584554266233380427062127, −16.18378316317082013426225272386, −15.19678608713427052055669967174, −14.34938555301357340613147664892, −13.56422008193368168569918154585, −12.85121830072179773420681244209, −11.774291866894622107150281782479, −10.69912134154496184842692106400, −9.7361500647772178534849701019, −9.02009333595259582060176556393, −8.47816935046240029387832235614, −7.49174239368447629260363110121, −6.468281442524377676492175718090, −5.45011409482181360026762411339, −5.22538011813521415393855245107, −3.98227833803782449423887377117, −2.1974260660450538121040368722, −1.556451586764664881046950686087, 0.31839955876334976891217443863, 1.73104887777299213574539801245, 2.55656771598374452953868212304, 3.47096035180467805535925369298, 4.49611105114725359302816303376, 5.52166175215318810682039529269, 6.747553503928060737928276480229, 7.65630560280647331657265590712, 8.40619749481787202893844781414, 9.48688877896045458543115169039, 10.423606936920464578464358137384, 10.70929133852134160564818386173, 11.429790169165618866584225157023, 12.99418372982269144124889484158, 13.13864712639177198042958003113, 14.03365009832439073547856866527, 15.01195588018377407004145174969, 16.06648115054316791384082010868, 17.19404127357554362146683839180, 17.65774553167546232835527091299, 18.27837594453004574729039179386, 19.06634599659094452050781010935, 20.14663971174786930725699900712, 20.52791187255711972778497436736, 21.39587919489744794899032076861

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