Properties

Label 2-99-1.1-c1-0-0
Degree $2$
Conductor $99$
Sign $1$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 4·7-s + 3·8-s − 2·10-s − 11-s − 2·13-s − 4·14-s − 16-s + 2·17-s − 2·20-s + 22-s − 8·23-s − 25-s + 2·26-s − 4·28-s + 6·29-s − 8·31-s − 5·32-s − 2·34-s + 8·35-s + 6·37-s + 6·40-s + 2·41-s + 44-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.06·8-s − 0.632·10-s − 0.301·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 1.35·35-s + 0.986·37-s + 0.948·40-s + 0.312·41-s + 0.150·44-s + 1.17·46-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7923074038\)
\(L(\frac12)\) \(\approx\) \(0.7923074038\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.12737143370045703342633485701, −12.97885466615636407297758144806, −11.61114553747330884538582571440, −10.39134307397473772625111872930, −9.637990407765945223554197990480, −8.389268452519734683124359977272, −7.60731726518247812897599595849, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −1.82001999803537272093437549398, 1.82001999803537272093437549398, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 7.60731726518247812897599595849, 8.389268452519734683124359977272, 9.637990407765945223554197990480, 10.39134307397473772625111872930, 11.61114553747330884538582571440, 12.97885466615636407297758144806, 14.12737143370045703342633485701

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