Properties

Degree 2
Conductor $ 3 \cdot 11 $
Sign $0.957 - 0.288i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 1.65i)3-s − 2·4-s − 3.31i·5-s + (−2.5 + 1.65i)9-s + 3.31i·11-s + (−1 − 3.31i)12-s + (5.5 − 1.65i)15-s + 4·16-s + 6.63i·20-s − 3.31i·23-s − 6·25-s + (−4 − 3.31i)27-s + 5·31-s + (−5.5 + 1.65i)33-s + (5 − 3.31i)36-s − 7·37-s + ⋯
L(s)  = 1  + (0.288 + 0.957i)3-s − 4-s − 1.48i·5-s + (−0.833 + 0.552i)9-s + 1.00i·11-s + (−0.288 − 0.957i)12-s + (1.42 − 0.428i)15-s + 16-s + 1.48i·20-s − 0.691i·23-s − 1.20·25-s + (−0.769 − 0.638i)27-s + 0.898·31-s + (−0.957 + 0.288i)33-s + (0.833 − 0.552i)36-s − 1.15·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33\)    =    \(3 \cdot 11\)
\( \varepsilon \)  =  $0.957 - 0.288i$
motivic weight  =  \(1\)
character  :  $\chi_{33} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 33,\ (\ :1/2),\ 0.957 - 0.288i)$
$L(1)$  $\approx$  $0.660751 + 0.0974455i$
$L(\frac12)$  $\approx$  $0.660751 + 0.0974455i$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 - 1.65i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + 3.31iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−16.85574838471113963425433979456, −15.74352154365985779192189445348, −14.48010355709446529915257600066, −13.25737799010645723889294192930, −12.18377057708802742812991424989, −10.14183943032400540896029256073, −9.160132383755029723980325860329, −8.264947596125555144370168258865, −5.18130267836277930636751198273, −4.27630887656717266868263341494, 3.25699087747366984275342802206, 6.01987084005356094215710711834, 7.48410396012403277235033293890, 8.841889407311114617556569512502, 10.50315208485772406638675869216, 11.91285142562008747901103505623, 13.52133580755889374274588661744, 14.04167765341748268011552322659, 15.15355040523506862802895652520, 17.17113961041653810289302193238

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