Properties

Label 2-99-33.20-c2-0-7
Degree $2$
Conductor $99$
Sign $-0.935 + 0.353i$
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  + (0.296 + 0.0962i)2-s + (−3.15 − 2.29i)4-s + (−5.65 + 1.83i)5-s + (−7.01 − 5.09i)7-s + (−1.44 − 1.99i)8-s − 1.85·10-s + (−0.122 + 10.9i)11-s + (5.77 − 17.7i)13-s + (−1.58 − 2.18i)14-s + (4.58 + 14.1i)16-s + (−9.12 + 2.96i)17-s + (−4.66 + 3.38i)19-s + (22.0 + 7.17i)20-s + (−1.09 + 3.24i)22-s − 41.5i·23-s + ⋯
L(s)  = 1  + (0.148 + 0.0481i)2-s + (−0.789 − 0.573i)4-s + (−1.13 + 0.367i)5-s + (−1.00 − 0.728i)7-s + (−0.180 − 0.248i)8-s − 0.185·10-s + (−0.0111 + 0.999i)11-s + (0.444 − 1.36i)13-s + (−0.113 − 0.156i)14-s + (0.286 + 0.882i)16-s + (−0.537 + 0.174i)17-s + (−0.245 + 0.178i)19-s + (1.10 + 0.358i)20-s + (−0.0497 + 0.147i)22-s − 1.80i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.935 + 0.353i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0517247 - 0.282936i\)
\(L(\frac12)\) \(\approx\) \(0.0517247 - 0.282936i\)
\(L(2)\) 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 \( 1 + (0.122 - 10.9i)T \)
good2 \( 1 + (-0.296 - 0.0962i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (5.65 - 1.83i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (7.01 + 5.09i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-5.77 + 17.7i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (9.12 - 2.96i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (4.66 - 3.38i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 41.5iT - 529T^{2} \)
29 \( 1 + (10.2 - 14.0i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (4.22 - 13.0i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-5.83 - 4.24i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (31.2 + 42.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 43.3T + 1.84e3T^{2} \)
47 \( 1 + (-11.0 - 15.2i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (51.8 + 16.8i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (20.7 - 28.5i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (36.4 + 112. i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 91.5T + 4.48e3T^{2} \)
71 \( 1 + (110. - 35.7i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-42.8 - 31.1i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-0.633 + 1.94i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-27.4 + 8.90i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 134. iT - 7.92e3T^{2} \)
97 \( 1 + (3.08 - 9.48i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.02849600351053334146908455952, −12.49896113833478465669766144292, −10.73695576722217363197441291469, −10.18092511691819921796975245391, −8.783363454220447165809835238587, −7.50103564204058271367275470483, −6.30691958059854367411072311599, −4.58578995448035567455754296320, −3.49421578362870922064437080243, −0.21015577802722281678265804073, 3.30782351185196318812943236596, 4.34243084683414706830181047132, 6.00817338809123284359021460028, 7.61233881926911881285062932985, 8.804067865191379715994022205158, 9.354287434792708608256151518049, 11.39987478292234460035207390386, 11.96069080284706868319009361289, 13.11979386950903217249163474518, 13.79036232397370166897857083439

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