Properties

Label 2-99-1.1-c7-0-6
Degree $2$
Conductor $99$
Sign $1$
Analytic cond. $30.9261$
Root an. cond. $5.56112$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 28·4-s + 410·5-s − 1.02e3·7-s + 1.56e3·8-s − 4.10e3·10-s + 1.33e3·11-s + 1.29e4·13-s + 1.02e4·14-s − 1.20e4·16-s − 1.70e4·17-s − 5.41e4·19-s − 1.14e4·20-s − 1.33e4·22-s + 1.14e4·23-s + 8.99e4·25-s − 1.29e5·26-s + 2.87e4·28-s + 1.86e5·29-s − 1.88e5·31-s − 7.95e4·32-s + 1.70e5·34-s − 4.21e5·35-s + 3.95e5·37-s + 5.41e5·38-s + 6.39e5·40-s + 4.75e4·41-s + ⋯
L(s)  = 1  − 0.883·2-s − 0.218·4-s + 1.46·5-s − 1.13·7-s + 1.07·8-s − 1.29·10-s + 0.301·11-s + 1.63·13-s + 1.00·14-s − 0.733·16-s − 0.842·17-s − 1.81·19-s − 0.320·20-s − 0.266·22-s + 0.196·23-s + 1.15·25-s − 1.44·26-s + 0.247·28-s + 1.42·29-s − 1.13·31-s − 0.428·32-s + 0.744·34-s − 1.66·35-s + 1.28·37-s + 1.60·38-s + 1.58·40-s + 0.107·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(30.9261\)
Root analytic conductor: \(5.56112\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.272065352\)
\(L(\frac12)\) \(\approx\) \(1.272065352\)
\(L(\frac{9}{2})\) 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 \( 1 - p^{3} T \)
good2 \( 1 + 5 p T + p^{7} T^{2} \)
5 \( 1 - 82 p T + p^{7} T^{2} \)
7 \( 1 + 1028 T + p^{7} T^{2} \)
13 \( 1 - 12958 T + p^{7} T^{2} \)
17 \( 1 + 17062 T + p^{7} T^{2} \)
19 \( 1 + 54168 T + p^{7} T^{2} \)
23 \( 1 - 11488 T + p^{7} T^{2} \)
29 \( 1 - 186654 T + p^{7} T^{2} \)
31 \( 1 + 188672 T + p^{7} T^{2} \)
37 \( 1 - 395886 T + p^{7} T^{2} \)
41 \( 1 - 47546 T + p^{7} T^{2} \)
43 \( 1 - 602088 T + p^{7} T^{2} \)
47 \( 1 - 647200 T + p^{7} T^{2} \)
53 \( 1 - 1312722 T + p^{7} T^{2} \)
59 \( 1 - 2681140 T + p^{7} T^{2} \)
61 \( 1 - 551190 T + p^{7} T^{2} \)
67 \( 1 - 459260 T + p^{7} T^{2} \)
71 \( 1 - 18072 T + p^{7} T^{2} \)
73 \( 1 + 426062 T + p^{7} T^{2} \)
79 \( 1 - 297764 T + p^{7} T^{2} \)
83 \( 1 + 5684028 T + p^{7} T^{2} \)
89 \( 1 - 6342966 T + p^{7} T^{2} \)
97 \( 1 - 16651586 T + p^{7} T^{2} \)
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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

−13.01508002567190870860294212158, −10.89229155542069898472499755955, −10.17502069329538658362793235159, −9.148554291427757468073240639227, −8.623199291408181836175869242133, −6.69895158553024582075554915765, −5.93347562149139051939729834623, −4.10809983599880512602002387262, −2.21231529065361730735993550838, −0.823253037998103865481322327784, 0.823253037998103865481322327784, 2.21231529065361730735993550838, 4.10809983599880512602002387262, 5.93347562149139051939729834623, 6.69895158553024582075554915765, 8.623199291408181836175869242133, 9.148554291427757468073240639227, 10.17502069329538658362793235159, 10.89229155542069898472499755955, 13.01508002567190870860294212158

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