Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.985 - 0.172i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.447 + 0.774i)2-s + (1.22 − 1.22i)3-s + (0.599 − 1.03i)4-s + (−1.87 + 3.24i)5-s + (1.49 + 0.401i)6-s + (−0.725 − 1.25i)7-s + 2.86·8-s + (0.00384 − 2.99i)9-s − 3.35·10-s + (−0.5 − 0.866i)11-s + (−0.536 − 2.00i)12-s + (−2.87 + 4.98i)13-s + (0.648 − 1.12i)14-s + (1.67 + 6.27i)15-s + (0.0800 + 0.138i)16-s − 4.79·17-s + ⋯
L(s)  = 1  + (0.316 + 0.547i)2-s + (0.707 − 0.706i)3-s + (0.299 − 0.519i)4-s + (−0.838 + 1.45i)5-s + (0.610 + 0.164i)6-s + (−0.274 − 0.474i)7-s + 1.01·8-s + (0.00128 − 0.999i)9-s − 1.06·10-s + (−0.150 − 0.261i)11-s + (−0.154 − 0.579i)12-s + (−0.798 + 1.38i)13-s + (0.173 − 0.300i)14-s + (0.432 + 1.61i)15-s + (0.0200 + 0.0346i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.985 - 0.172i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 99,\ (\ :1/2),\ 0.985 - 0.172i)\)
\(L(1)\)  \(\approx\)  \(1.31630 + 0.114312i\)
\(L(\frac12)\)  \(\approx\)  \(1.31630 + 0.114312i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.447 - 0.774i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.87 - 3.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.725 + 1.25i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.87 - 4.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
19 \( 1 - 0.702T + 19T^{2} \)
23 \( 1 + (-0.825 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 - 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.65 + 2.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (2.12 - 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.05 - 3.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.898 + 1.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.27 + 2.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.47 - 7.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.14T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + (-0.543 - 0.941i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.90 - 3.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.01T + 89T^{2} \)
97 \( 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.22609111974396112909102139623, −13.30563508595311377656147436803, −11.71717730941117642097035276205, −10.89851348719835940624499416111, −9.612619986259039467935072649630, −7.912919709698364022535120311952, −6.87426322366561269074669448843, −6.58999612750236329830392779345, −4.20396939852830988930132799869, −2.53264539477607638353543442603, 2.68795680624850277873737133833, 4.13303628190339915877173735442, 5.07094737039958634117753956355, 7.63944406405573174015519526584, 8.374108585475706663460734330347, 9.463464412505869606654904543640, 10.78875164902541139869842643641, 12.00881297090682081789954659508, 12.74395279961215987922752073318, 13.48744339972181298404034992029

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