↑

Properties

Label 2-1911-273.59-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.467 - 0.884i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
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  + (−0.866 + 0.5i)3-s + i·4-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (0.5 + 1.86i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s − 0.999·39-s + (−1.5 + 0.866i)43-s + (0.866 − 0.5i)48-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + i·4-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (0.5 + 1.86i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s − 0.999·39-s + (−1.5 + 0.866i)43-s + (0.866 − 0.5i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.467 - 0.884i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.467 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8123256506\)
\(L(\frac12)\) \(\approx\) \(0.8123256506\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
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

−9.685298583833672961768219043033, −8.881382888933948832748228344449, −8.106730529002275246911142599364, −7.27436374018019618815334224022, −6.41945846990944440159541306570, −5.71489100699258092129151667464, −4.67451678981199043790295939944, −3.85786934908550481843425890460, −3.23037942181601737640970932826, −1.57225992738321893978378720057, 0.71442818763940620387889592891, 1.78754903395048475143507203624, 3.09378882939776712394218104537, 4.63822104995124372266777973811, 5.15529970829613517307417014940, 5.95072857628456377137976652811, 6.74180743114563932973195375333, 7.21617999341327136745042377698, 8.474595617694178730739169548557, 9.120303596634148470757959987088

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