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Properties

Label 2-99-11.3-c1-0-1
Degree $2$
Conductor $99$
Sign $0.530 - 0.847i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.224i)2-s + (−0.572 + 1.76i)4-s + (1.30 + 0.951i)5-s + (−0.309 + 0.951i)7-s + (−0.454 − 1.40i)8-s − 0.618·10-s + (2.19 + 2.48i)11-s + (3.42 − 2.48i)13-s + (−0.118 − 0.363i)14-s + (−2.54 − 1.84i)16-s + (−6.35 − 4.61i)17-s + (−0.263 − 0.812i)19-s + (−2.42 + 1.76i)20-s + (−1.23 − 0.277i)22-s + 4.23·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.158i)2-s + (−0.286 + 0.881i)4-s + (0.585 + 0.425i)5-s + (−0.116 + 0.359i)7-s + (−0.160 − 0.495i)8-s − 0.195·10-s + (0.660 + 0.750i)11-s + (0.950 − 0.690i)13-s + (−0.0315 − 0.0970i)14-s + (−0.636 − 0.462i)16-s + (−1.54 − 1.11i)17-s + (−0.0605 − 0.186i)19-s + (−0.542 + 0.394i)20-s + (−0.263 − 0.0591i)22-s + 0.883·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809464 + 0.448545i\)
\(L(\frac12)\) \(\approx\) \(0.809464 + 0.448545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 2.48i)T \)
good2 \( 1 + (0.309 - 0.224i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.42 + 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.35 + 4.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.11 - 2.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.545 - 1.67i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.30 - 4.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 + (0.336 + 1.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.11 - 1.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.97 - 9.14i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.92 + 5.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + (4.30 + 3.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.89 - 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.04 + 4.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.76T + 89T^{2} \)
97 \( 1 + (0.927 - 0.673i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.85404142267583260855248494458, −13.12589318711766496610486109219, −12.04727202776054269509203972594, −10.90290839657879756723029561217, −9.475806944679677370225573928854, −8.748218401930449856053425731779, −7.31212599076008857144553093446, −6.27743133691307765662514150910, −4.44893809068220476898388007449, −2.75797128229835304904206030614, 1.54716083591508772365809195642, 4.12309431720945309107310287530, 5.68557377996537153024881361506, 6.64230974815025429215943007626, 8.783123492788275939600923885725, 9.159105410768445855660326247727, 10.64593736831630152959543672430, 11.23293497331166116966124842986, 12.97189447599362422994810679148, 13.70470492580803328094639729031

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