↑

Properties

Label 2-1127-161.114-c0-0-8
Degree $2$
Conductor $1127$
Sign $-0.266 + 0.963i$
Analytic cond. $0.562446$
Root an. cond. $0.749964$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s − 0.999·6-s + 8-s + 13-s + (0.5 − 0.866i)16-s + (−0.5 + 0.866i)23-s + (−0.500 − 0.866i)24-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)39-s + 41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s − 0.999·6-s + 8-s + 13-s + (0.5 − 0.866i)16-s + (−0.5 + 0.866i)23-s + (−0.500 − 0.866i)24-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312089140\)
\(L(\frac12)\) \(\approx\) \(1.312089140\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02929629718358285413206972134, −9.066800266106310562839259573551, −7.85323380701671271106028131465, −7.39183160453245654258911734337, −6.28590637520615007201096102420, −5.64327155241069703588362568038, −4.25375539432778622411418137841, −3.57607872202454958612393336409, −2.26772748487487028694225720026, −1.28408681354923778933539869819, 1.79034456063027282311271813016, 3.63667066355729846682429716603, 4.38397673477583899584194832421, 5.31880029653584820588693884326, 5.83410849858096410964830666373, 6.75089436009230286956591724389, 7.61004858035604702656352035655, 8.523310950768372098375891448168, 9.564682515641307364691732435632, 10.34341293884542141807261731112

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