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Properties

Label 2-99-99.43-c0-0-0
Degree $2$
Conductor $99$
Sign $0.766 + 0.642i$
Analytic cond. $0.0494074$
Root an. cond. $0.222277$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.499 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (0.499 − 0.866i)20-s + (−1 − 1.73i)23-s + 0.999·27-s + (0.5 + 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.499 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (0.499 − 0.866i)20-s + (−1 − 1.73i)23-s + 0.999·27-s + (0.5 + 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.0494074\)
Root analytic conductor: \(0.222277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4774195106\)
\(L(\frac12)\) \(\approx\) \(0.4774195106\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−14.08413891845906402596993050808, −13.09202432732099513747878414976, −12.03143578339260485688789095758, −10.56827862765709099156355649704, −10.16722206896139673314314510775, −8.495025006313518922315157726615, −6.97978145828428605996788480269, −6.13008093900409168117416047022, −4.86750891631774348547014659670, −2.19749917619495695658025058344, 3.48574567035271635281071490245, 4.83958004195883381465049249391, 5.87745731392387153437427664513, 7.900036263668485527728149050920, 8.995618197728112279657030086600, 9.771058824857733952186506670259, 11.18660620224420896302432564705, 12.17300394541636825118828706018, 13.21712899441185516945422273870, 14.04537565277773264555820701746

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