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Properties

Label 2-99-33.29-c1-0-2
Degree $2$
Conductor $99$
Sign $0.538 - 0.842i$
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  + (1.97 + 1.43i)2-s + (1.21 + 3.74i)4-s + (−2.23 − 3.07i)5-s + (−0.349 + 0.113i)7-s + (−1.45 + 4.48i)8-s − 9.24i·10-s + (−2.97 + 1.46i)11-s + (−0.557 + 0.767i)13-s + (−0.852 − 0.276i)14-s + (−2.92 + 2.12i)16-s + (2.77 − 2.01i)17-s + (4.05 + 1.31i)19-s + (8.78 − 12.0i)20-s + (−7.96 − 1.38i)22-s + 4.96i·23-s + ⋯
L(s)  = 1  + (1.39 + 1.01i)2-s + (0.608 + 1.87i)4-s + (−0.997 − 1.37i)5-s + (−0.132 + 0.0429i)7-s + (−0.515 + 1.58i)8-s − 2.92i·10-s + (−0.897 + 0.440i)11-s + (−0.154 + 0.212i)13-s + (−0.227 − 0.0740i)14-s + (−0.731 + 0.531i)16-s + (0.673 − 0.489i)17-s + (0.929 + 0.302i)19-s + (1.96 − 2.70i)20-s + (−1.69 − 0.295i)22-s + 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46566 + 0.802662i\)
\(L(\frac12)\) \(\approx\) \(1.46566 + 0.802662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.97 - 1.46i)T \)
good2 \( 1 + (-1.97 - 1.43i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.23 + 3.07i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.349 - 0.113i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.557 - 0.767i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.77 + 2.01i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.05 - 1.31i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (0.767 + 2.36i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.84 + 2.06i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.21 + 6.83i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.840 - 2.58i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 + (-0.0195 - 0.00636i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.25 - 4.48i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-6.29 + 2.04i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.73 + 7.88i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 + (-6.06 - 8.35i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.18 - 1.35i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.42 + 8.84i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.42 + 5.39i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 3.04iT - 89T^{2} \)
97 \( 1 + (-12.2 - 8.86i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.04201293063829271846705432338, −13.03813541323763002464150551377, −12.38168053627552220007785988241, −11.57042096152934372678284536576, −9.476720488653930269262171637111, −7.949321529781420749180256409494, −7.40857621173943986376563249876, −5.56842413950146169869356768211, −4.81537837965316832944728771851, −3.58679690427864680825003633188, 2.82530119061808492440557214047, 3.62353736683087917681730009855, 5.18838066377936281132519083364, 6.62265673296252931427470162594, 7.951072966986249634631017642218, 10.20209394294633044986537921576, 10.82769839767526614585777184121, 11.68778495820493892370582099309, 12.59123865584960528392745061526, 13.70036399663193452124532460368

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