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Properties

Label 2-99-11.6-c2-0-3
Degree $2$
Conductor $99$
Sign $0.985 - 0.167i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.35 + 0.764i)2-s + (1.72 − 1.25i)4-s + (0.789 − 2.42i)5-s + (−0.100 − 0.137i)7-s + (2.72 − 3.74i)8-s + 6.32i·10-s + (7.69 + 7.85i)11-s + (18.3 − 5.95i)13-s + (0.341 + 0.247i)14-s + (−6.17 + 19.0i)16-s + (19.3 + 6.27i)17-s + (8.57 − 11.8i)19-s + (−1.67 − 5.16i)20-s + (−24.1 − 12.6i)22-s − 7.74·23-s + ⋯
L(s)  = 1  + (−1.17 + 0.382i)2-s + (0.430 − 0.312i)4-s + (0.157 − 0.485i)5-s + (−0.0143 − 0.0196i)7-s + (0.340 − 0.468i)8-s + 0.632i·10-s + (0.699 + 0.714i)11-s + (1.41 − 0.458i)13-s + (0.0243 + 0.0177i)14-s + (−0.385 + 1.18i)16-s + (1.13 + 0.368i)17-s + (0.451 − 0.621i)19-s + (−0.0839 − 0.258i)20-s + (−1.09 − 0.573i)22-s − 0.336·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.792016 + 0.0668940i\)
\(L(\frac12)\) \(\approx\) \(0.792016 + 0.0668940i\)
\(L(2)\) 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 \( 1 + (-7.69 - 7.85i)T \)
good2 \( 1 + (2.35 - 0.764i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (-0.789 + 2.42i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (0.100 + 0.137i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-18.3 + 5.95i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (-19.3 - 6.27i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-8.57 + 11.8i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 7.74T + 529T^{2} \)
29 \( 1 + (22.4 + 30.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (13.0 + 40.2i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (41.7 - 30.3i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-27.1 + 37.4i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 59.7iT - 1.84e3T^{2} \)
47 \( 1 + (-27.6 - 20.0i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-4.70 - 14.4i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (21.3 - 15.4i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-60.2 - 19.5i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 2.91T + 4.48e3T^{2} \)
71 \( 1 + (29.9 - 92.0i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (10.1 + 14.0i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (50.1 - 16.3i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (22.3 + 7.25i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 97.1T + 7.92e3T^{2} \)
97 \( 1 + (15.6 + 48.1i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−13.58157651267315284006594803481, −12.69710706006766523012620626989, −11.33377158692085894043215877113, −10.07564907479612078504887360151, −9.234789006674917617144942701342, −8.288353906007462156353269416150, −7.22648128940543809421227852275, −5.82797880185077867804028263367, −3.96690496061393265236403131831, −1.20806754230974581036079904622, 1.36053471595527203617195423625, 3.46526597350117810330318401219, 5.64455512861173609161813204274, 7.09385326170868981332656504772, 8.454456864903732172835985927261, 9.216353665338679599701197637711, 10.43126502958406453018265987197, 11.11727730911732704964758046131, 12.22334183435244542996502387666, 13.95074952859252746144809307694

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