Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $-0.357 + 0.934i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 2.13i)2-s + (1.66 + 0.461i)3-s + (−2.04 + 3.54i)4-s + (1.21 − 2.10i)5-s + (−1.07 − 4.14i)6-s + (−1.16 − 2.02i)7-s + 5.17·8-s + (2.57 + 1.54i)9-s − 6.01·10-s + (−0.5 − 0.866i)11-s + (−5.05 + 4.97i)12-s + (−2.35 + 4.07i)13-s + (−2.88 + 5.00i)14-s + (3.00 − 2.95i)15-s + (−2.29 − 3.97i)16-s − 3.20·17-s + ⋯
L(s)  = 1  + (−0.873 − 1.51i)2-s + (0.963 + 0.266i)3-s + (−1.02 + 1.77i)4-s + (0.544 − 0.943i)5-s + (−0.438 − 1.69i)6-s + (−0.441 − 0.765i)7-s + 1.83·8-s + (0.857 + 0.514i)9-s − 1.90·10-s + (−0.150 − 0.261i)11-s + (−1.46 + 1.43i)12-s + (−0.653 + 1.13i)13-s + (−0.771 + 1.33i)14-s + (0.776 − 0.764i)15-s + (−0.574 − 0.994i)16-s − 0.778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $-0.357 + 0.934i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 99,\ (\ :1/2),\ -0.357 + 0.934i)\)
\(L(1)\)  \(\approx\)  \(0.490556 - 0.712859i\)
\(L(\frac12)\)  \(\approx\)  \(0.490556 - 0.712859i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.66 - 0.461i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.21 + 2.10i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.16 + 2.02i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.35 - 4.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 - 7.77T + 19T^{2} \)
23 \( 1 + (1.37 - 2.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.18 - 2.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.685 + 1.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + (1.77 - 3.07i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.103 + 0.180i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.11T + 53T^{2} \)
59 \( 1 + (-0.120 + 0.208i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.830 + 1.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.84 + 6.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (6.35 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.25 + 9.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (4.46 + 7.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.42454191780361585691677259630, −12.48319203581210370432692514123, −11.29548326812780193307231310377, −9.972933521363556965723913695087, −9.476971407994441382397427511475, −8.680484780498553269733903101776, −7.36475084377932927142074433104, −4.62671001098699171328357433170, −3.25578403595796048649443444071, −1.62424820920486031135675820796, 2.75932399467404698268855279009, 5.45880356007829472884828902541, 6.68449967536954070543052611740, 7.48286579933188513050371731775, 8.587047621713204809614361172403, 9.607742392770992060020545825915, 10.27472333327391457412285759835, 12.42239003264907429974602038613, 13.81389601399309085656520003739, 14.41716779824769410993948642092

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