Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.766 - 0.642i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1 + 1.73i)23-s + 0.999·27-s + (0.5 − 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1 + 1.73i)23-s + 0.999·27-s + (0.5 − 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.766 - 0.642i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 99,\ (\ :0),\ 0.766 - 0.642i)$
$L(\frac{1}{2})$  $\approx$  $0.4774195106$
$L(\frac12)$  $\approx$  $0.4774195106$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−14.04537565277773264555820701746, −13.21712899441185516945422273870, −12.17300394541636825118828706018, −11.18660620224420896302432564705, −9.771058824857733952186506670259, −8.995618197728112279657030086600, −7.900036263668485527728149050920, −5.87745731392387153437427664513, −4.83958004195883381465049249391, −3.48574567035271635281071490245, 2.19749917619495695658025058344, 4.86750891631774348547014659670, 6.13008093900409168117416047022, 6.97978145828428605996788480269, 8.495025006313518922315157726615, 10.16722206896139673314314510775, 10.56827862765709099156355649704, 12.03143578339260485688789095758, 13.09202432732099513747878414976, 14.08413891845906402596993050808

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