Properties

Degree $2$
Conductor $99$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 31·4-s + 92·5-s − 26·7-s + 63·8-s − 92·10-s − 121·11-s − 692·13-s + 26·14-s + 929·16-s + 1.44e3·17-s + 2.16e3·19-s − 2.85e3·20-s + 121·22-s + 1.58e3·23-s + 5.33e3·25-s + 692·26-s + 806·28-s + 5.52e3·29-s + 4.79e3·31-s − 2.94e3·32-s − 1.44e3·34-s − 2.39e3·35-s − 1.01e4·37-s − 2.16e3·38-s + 5.79e3·40-s + 1.06e4·41-s + ⋯
L(s)  = 1  − 0.176·2-s − 0.968·4-s + 1.64·5-s − 0.200·7-s + 0.348·8-s − 0.290·10-s − 0.301·11-s − 1.13·13-s + 0.0354·14-s + 0.907·16-s + 1.21·17-s + 1.37·19-s − 1.59·20-s + 0.0533·22-s + 0.623·23-s + 1.70·25-s + 0.200·26-s + 0.194·28-s + 1.22·29-s + 0.895·31-s − 0.508·32-s − 0.213·34-s − 0.330·35-s − 1.22·37-s − 0.242·38-s + 0.572·40-s + 0.986·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(5\)
Character: $\chi_{99} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.76514\)
\(L(\frac12)\) \(\approx\) \(1.76514\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + p^{2} T \)
good2 \( 1 + T + p^{5} T^{2} \)
5 \( 1 - 92 T + p^{5} T^{2} \)
7 \( 1 + 26 T + p^{5} T^{2} \)
13 \( 1 + 692 T + p^{5} T^{2} \)
17 \( 1 - 1442 T + p^{5} T^{2} \)
19 \( 1 - 2160 T + p^{5} T^{2} \)
23 \( 1 - 1582 T + p^{5} T^{2} \)
29 \( 1 - 5526 T + p^{5} T^{2} \)
31 \( 1 - 4792 T + p^{5} T^{2} \)
37 \( 1 + 10194 T + p^{5} T^{2} \)
41 \( 1 - 10622 T + p^{5} T^{2} \)
43 \( 1 - 8580 T + p^{5} T^{2} \)
47 \( 1 - 2362 T + p^{5} T^{2} \)
53 \( 1 - 30804 T + p^{5} T^{2} \)
59 \( 1 + 6416 T + p^{5} T^{2} \)
61 \( 1 - 42096 T + p^{5} T^{2} \)
67 \( 1 + 28444 T + p^{5} T^{2} \)
71 \( 1 + 45690 T + p^{5} T^{2} \)
73 \( 1 + 18374 T + p^{5} T^{2} \)
79 \( 1 + 105214 T + p^{5} T^{2} \)
83 \( 1 + 62292 T + p^{5} T^{2} \)
89 \( 1 - 72246 T + p^{5} T^{2} \)
97 \( 1 - 79262 T + p^{5} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.12084720029171735598519572642, −12.14443880496896017704111034409, −10.13695892938027687712688929295, −9.897198760689012823339660857853, −8.826671232216666805418971513375, −7.36084609443185827375893685154, −5.73677102048045892481498855565, −4.93640382348827436648231630641, −2.86622213204796619008894915820, −1.06700858021226337619949574352, 1.06700858021226337619949574352, 2.86622213204796619008894915820, 4.93640382348827436648231630641, 5.73677102048045892481498855565, 7.36084609443185827375893685154, 8.826671232216666805418971513375, 9.897198760689012823339660857853, 10.13695892938027687712688929295, 12.14443880496896017704111034409, 13.12084720029171735598519572642

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