Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.751 - 0.659i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 + 0.951i)2-s + (0.190 + 0.587i)4-s + (−0.309 + 0.224i)5-s + (0.927 + 2.85i)7-s + (0.690 − 2.12i)8-s − 0.618·10-s + (−2.80 − 1.76i)11-s + (−5.04 − 3.66i)13-s + (−1.5 + 4.61i)14-s + (3.92 − 2.85i)16-s + (−0.5 + 0.363i)17-s + (−0.263 + 0.812i)19-s + (−0.190 − 0.138i)20-s + (−2 − 4.97i)22-s + 5.47·23-s + ⋯
L(s)  = 1  + (0.925 + 0.672i)2-s + (0.0954 + 0.293i)4-s + (−0.138 + 0.100i)5-s + (0.350 + 1.07i)7-s + (0.244 − 0.751i)8-s − 0.195·10-s + (−0.846 − 0.531i)11-s + (−1.39 − 1.01i)13-s + (−0.400 + 1.23i)14-s + (0.981 − 0.713i)16-s + (−0.121 + 0.0881i)17-s + (−0.0605 + 0.186i)19-s + (−0.0427 − 0.0310i)20-s + (−0.426 − 1.06i)22-s + 1.14·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.751 - 0.659i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 99,\ (\ :1/2),\ 0.751 - 0.659i)$
$L(1)$  $\approx$  $1.36063 + 0.512020i$
$L(\frac12)$  $\approx$  $1.36063 + 0.512020i$
$L(\frac{3}{2})$   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 \( 1 \)
11 \( 1 + (2.80 + 1.76i)T \)
good2 \( 1 + (-1.30 - 0.951i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.309 - 0.224i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.927 - 2.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.04 + 3.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.11 - 2.26i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.30 + 4.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.83 - 5.65i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 + (-0.190 + 0.587i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.97 + 4.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.64 - 5.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.927 - 0.673i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (-11.7 + 8.55i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.381 - 1.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.427 + 0.310i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.2 - 7.46i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 + (12.1 + 8.83i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.21849421328805753949467842849, −12.99581120630903484402693634626, −12.36057736510634857375458834021, −10.94532583001237446605104749711, −9.706824278428610577373155902898, −8.250019655275872751043527251637, −7.08585544143943675140507218754, −5.57709364750440716429837695488, −5.00212064571773703624658333110, −2.99109732830900938047165688224, 2.44875023367782687263432536837, 4.24605824971914652230569860549, 4.96092440515681747857783805520, 7.03175841904256586093581601287, 8.077741216665055473900571049257, 9.796024811089212381660377861908, 10.85663087061310372691406761102, 11.84797977893248488195440637564, 12.72604380439538797615311341720, 13.72142918954134949324861283565

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