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Properties

Label 2-99-99.32-c1-0-9
Degree $2$
Conductor $99$
Sign $-0.283 + 0.959i$
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  + (−1.68 − 0.396i)3-s + (1 − 1.73i)4-s + (−3.68 − 2.12i)5-s + (2.68 + 1.33i)9-s + (2.87 − 1.65i)11-s + (−2.37 + 2.52i)12-s + (5.37 + 5.04i)15-s + (−1.99 − 3.46i)16-s + (−7.37 + 4.25i)20-s + (2.87 + 1.65i)23-s + (6.55 + 11.3i)25-s + (−4 − 3.31i)27-s + (5.55 − 9.62i)31-s + (−5.5 + 1.65i)33-s + (5 − 3.31i)36-s − 5.11·37-s + ⋯
L(s)  = 1  + (−0.973 − 0.228i)3-s + (0.5 − 0.866i)4-s + (−1.64 − 0.951i)5-s + (0.895 + 0.445i)9-s + (0.866 − 0.500i)11-s + (−0.684 + 0.728i)12-s + (1.38 + 1.30i)15-s + (−0.499 − 0.866i)16-s + (−1.64 + 0.951i)20-s + (0.598 + 0.345i)23-s + (1.31 + 2.27i)25-s + (−0.769 − 0.638i)27-s + (0.998 − 1.72i)31-s + (−0.957 + 0.288i)33-s + (0.833 − 0.552i)36-s − 0.841·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.283 + 0.959i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.283 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382536 - 0.511743i\)
\(L(\frac12)\) \(\approx\) \(0.382536 - 0.511743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 + 0.396i)T \)
11 \( 1 + (-2.87 + 1.65i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (3.68 + 2.12i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.87 - 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.55 + 9.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.11T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.12 + 3.53i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.43iT - 53T^{2} \)
59 \( 1 + (-9.81 - 5.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.69iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (-8.55 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.40273926519804500983737521302, −12.08750179933281686815598982503, −11.62897152922342046564522499400, −10.78494174959568348811470715475, −9.264904265355452950552686169486, −7.83787309733772378715696128377, −6.69766828677224253043623987357, −5.36900714361110054154701414779, −4.15951744692698623530818360822, −0.907798973200438474309634526588, 3.37289532433345097135088664317, 4.44231869870708802919807151731, 6.70879662021436986875811298569, 7.15342383655176570888930400320, 8.492941682877644108538145337693, 10.37901270499909754943815970642, 11.30661199998359670653171897410, 11.91724469874273935706154947141, 12.59591375238514878256494960649, 14.51579993206722124459419563777

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