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Properties

Label 2-99-9.7-c3-0-15
Degree $2$
Conductor $99$
Sign $0.997 + 0.0688i$
Analytic cond. $5.84118$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.09 + 3.62i)2-s + (−3.47 + 3.86i)3-s + (−4.73 − 8.20i)4-s + (−2.12 − 3.68i)5-s + (−6.72 − 20.6i)6-s + (10.7 − 18.5i)7-s + 6.17·8-s + (−2.84 − 26.8i)9-s + 17.7·10-s + (−5.5 + 9.52i)11-s + (48.1 + 10.2i)12-s + (−1.88 − 3.26i)13-s + (44.8 + 77.6i)14-s + (21.6 + 4.58i)15-s + (24.9 − 43.2i)16-s − 37.6·17-s + ⋯
L(s)  = 1  + (−0.739 + 1.28i)2-s + (−0.668 + 0.743i)3-s + (−0.592 − 1.02i)4-s + (−0.190 − 0.329i)5-s + (−0.457 − 1.40i)6-s + (0.578 − 1.00i)7-s + 0.273·8-s + (−0.105 − 0.994i)9-s + 0.562·10-s + (−0.150 + 0.261i)11-s + (1.15 + 0.245i)12-s + (−0.0402 − 0.0696i)13-s + (0.855 + 1.48i)14-s + (0.372 + 0.0789i)15-s + (0.390 − 0.676i)16-s − 0.537·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0688i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.997 + 0.0688i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.997 + 0.0688i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.499521 - 0.0172286i\)
\(L(\frac12)\) \(\approx\) \(0.499521 - 0.0172286i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.47 - 3.86i)T \)
11 \( 1 + (5.5 - 9.52i)T \)
good2 \( 1 + (2.09 - 3.62i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (2.12 + 3.68i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-10.7 + 18.5i)T + (-171.5 - 297. i)T^{2} \)
13 \( 1 + (1.88 + 3.26i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 37.6T + 4.91e3T^{2} \)
19 \( 1 + 9.79T + 6.85e3T^{2} \)
23 \( 1 + (-37.3 - 64.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-93.6 + 162. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (143. + 248. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 199.T + 5.06e4T^{2} \)
41 \( 1 + (100. + 173. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-120. + 208. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-148. + 256. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 385.T + 1.48e5T^{2} \)
59 \( 1 + (133. + 230. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (139. - 241. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-307. - 532. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 440.T + 3.57e5T^{2} \)
73 \( 1 - 964.T + 3.89e5T^{2} \)
79 \( 1 + (-424. + 734. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-367. + 636. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + (812. - 1.40e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.76440460359783083951133372263, −12.18524949532534141521852028849, −10.97780382640830923636307472894, −9.974420341289110190905927967102, −8.891802044546763929530507925998, −7.74067237077604479506645514470, −6.68589875437353699812737989122, −5.37242619329400378179643152760, −4.18419740774419881166716531665, −0.41357689563963836788012194765, 1.50848542694355964560503789738, 2.85259597257509705920701293042, 5.16819777009633839165828043475, 6.66697301633363810102722697143, 8.202419752107733904762696376860, 9.099057780326105917213827579750, 10.74152766417558128768942693174, 11.11842683150318268422338195008, 12.20403642038597881355672118762, 12.74515159883549948933286016787

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