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

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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  :  $\chi_{99} (34, \cdot )$
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 \( 1 + (1.64 - 2.85i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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