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Properties

Label 2-99-33.17-c1-0-0
Degree $2$
Conductor $99$
Sign $-0.336 - 0.941i$
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  + (0.726 + 2.23i)2-s + (−2.85 + 2.07i)4-s + (2.13 + 0.694i)5-s + (−2.38 − 3.27i)7-s + (−2.90 − 2.10i)8-s + 5.28i·10-s + (−3.31 − 0.0200i)11-s + (4.42 − 1.43i)13-s + (5.59 − 7.70i)14-s + (0.427 − 1.31i)16-s + (−0.0235 + 0.0725i)17-s + (1.40 − 1.93i)19-s + (−7.53 + 2.44i)20-s + (−2.36 − 7.42i)22-s + 3.22i·23-s + ⋯
L(s)  = 1  + (0.513 + 1.58i)2-s + (−1.42 + 1.03i)4-s + (0.956 + 0.310i)5-s + (−0.899 − 1.23i)7-s + (−1.02 − 0.745i)8-s + 1.67i·10-s + (−0.999 − 0.00604i)11-s + (1.22 − 0.398i)13-s + (1.49 − 2.05i)14-s + (0.106 − 0.328i)16-s + (−0.00571 + 0.0175i)17-s + (0.323 − 0.444i)19-s + (−1.68 + 0.547i)20-s + (−0.504 − 1.58i)22-s + 0.672i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.336 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727260 + 1.03189i\)
\(L(\frac12)\) \(\approx\) \(0.727260 + 1.03189i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.0200i)T \)
good2 \( 1 + (-0.726 - 2.23i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-2.13 - 0.694i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.38 + 3.27i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.42 + 1.43i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0235 - 0.0725i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.40 + 1.93i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.22iT - 23T^{2} \)
29 \( 1 + (-1.48 + 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.517 - 1.59i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.87 - 4.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.82 + 4.96i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.28iT - 43T^{2} \)
47 \( 1 + (3.65 - 5.02i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.16 + 0.379i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.341 + 0.469i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.59 + 1.16i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (-1.06 - 0.346i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.82 - 10.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.627 + 0.203i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.15 - 9.71i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (-5.08 - 15.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.88994044310061544198249999240, −13.67070770299610200835126909108, −12.90429995669593127185521975189, −10.75435238436121644016351947946, −9.832554752704275933229372628553, −8.347092907979979525834824217200, −7.14524696935193130942184018320, −6.31637085110312267711094418200, −5.25761067957580179302429516975, −3.56752487493087648862174388482, 2.03259686727371635080682113062, 3.30445251546400004271626914298, 5.15290558775792098491728666425, 6.15994412503645331716174729781, 8.662282480687396701513246222127, 9.596078100400562670761207510324, 10.42273505605839528619839274634, 11.61532813923100544107823831231, 12.63573314398045267518480083515, 13.20064895189009404318232394813

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