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Properties

Label 2-99-11.2-c2-0-1
Degree $2$
Conductor $99$
Sign $0.304 - 0.952i$
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  + (−0.625 − 0.203i)2-s + (−2.88 − 2.09i)4-s + (2.50 + 7.70i)5-s + (−2.40 + 3.30i)7-s + (2.92 + 4.02i)8-s − 5.32i·10-s + (6.70 + 8.72i)11-s + (14.0 + 4.56i)13-s + (2.17 − 1.57i)14-s + (3.39 + 10.4i)16-s + (−28.0 + 9.09i)17-s + (−7.52 − 10.3i)19-s + (8.93 − 27.4i)20-s + (−2.41 − 6.81i)22-s + 12.6·23-s + ⋯
L(s)  = 1  + (−0.312 − 0.101i)2-s + (−0.721 − 0.524i)4-s + (0.500 + 1.54i)5-s + (−0.343 + 0.472i)7-s + (0.365 + 0.503i)8-s − 0.532i·10-s + (0.609 + 0.792i)11-s + (1.08 + 0.351i)13-s + (0.155 − 0.112i)14-s + (0.212 + 0.653i)16-s + (−1.64 + 0.535i)17-s + (−0.395 − 0.544i)19-s + (0.446 − 1.37i)20-s + (−0.109 − 0.309i)22-s + 0.549·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.772858 + 0.564362i\)
\(L(\frac12)\) \(\approx\) \(0.772858 + 0.564362i\)
\(L(2)\) 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 \( 1 + (-6.70 - 8.72i)T \)
good2 \( 1 + (0.625 + 0.203i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-2.50 - 7.70i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (2.40 - 3.30i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-14.0 - 4.56i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (28.0 - 9.09i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (7.52 + 10.3i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 12.6T + 529T^{2} \)
29 \( 1 + (-7.85 + 10.8i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-10.9 + 33.8i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-14.5 - 10.5i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (9.51 + 13.0i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 44.2iT - 1.84e3T^{2} \)
47 \( 1 + (7.76 - 5.64i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (17.8 - 54.9i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-81.1 - 58.9i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-79.0 + 25.6i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 87.5T + 4.48e3T^{2} \)
71 \( 1 + (21.1 + 65.1i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (15.2 - 21.0i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (62.3 + 20.2i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-83.0 + 26.9i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 29.3T + 7.92e3T^{2} \)
97 \( 1 + (11.3 - 34.9i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.86149538185838605094299787894, −13.12473444544959254001877816561, −11.36396177322057915458300318485, −10.60864720746667275840301837126, −9.577196166183054895345946761275, −8.691753722152722118047967985586, −6.82850821061805915882284500525, −6.05588784464561404735027447710, −4.17679033337462435754732855611, −2.22416464325518925981549369235, 0.886859301183023887512532821716, 3.82444869354517569087513652328, 5.02799438692602511213685996930, 6.58574077940279245213696380563, 8.467030744772085662227631190829, 8.758732433454651916307231560024, 9.870234332931733127400383324452, 11.36351299881591626282317150985, 12.85833573398044012435070635264, 13.18824890728826606349974525613

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