Properties

Label 2-99-33.8-c7-0-21
Degree $2$
Conductor $99$
Sign $0.00774 + 0.999i$
Analytic cond. $30.9261$
Root an. cond. $5.56112$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.51 − 1.09i)2-s + (−38.4 + 118. i)4-s + (76.1 − 104. i)5-s + (−748. − 243. i)7-s + (145. + 448. i)8-s − 241. i·10-s + (2.32e3 + 3.75e3i)11-s + (−2.63e3 − 3.62e3i)13-s + (−1.39e3 + 454. i)14-s + (−1.21e4 − 8.84e3i)16-s + (−1.62e4 − 1.18e4i)17-s + (1.72e4 − 5.61e3i)19-s + (9.48e3 + 1.30e4i)20-s + (7.63e3 + 3.12e3i)22-s − 1.06e4i·23-s + ⋯
L(s)  = 1  + (0.133 − 0.0970i)2-s + (−0.300 + 0.925i)4-s + (0.272 − 0.374i)5-s + (−0.824 − 0.268i)7-s + (0.100 + 0.309i)8-s − 0.0765i·10-s + (0.525 + 0.850i)11-s + (−0.332 − 0.457i)13-s + (−0.136 + 0.0442i)14-s + (−0.743 − 0.540i)16-s + (−0.804 − 0.584i)17-s + (0.578 − 0.187i)19-s + (0.264 + 0.364i)20-s + (0.152 + 0.0625i)22-s − 0.181i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00774 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.00774 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.00774 + 0.999i$
Analytic conductor: \(30.9261\)
Root analytic conductor: \(5.56112\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :7/2),\ 0.00774 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.776793 - 0.770802i\)
\(L(\frac12)\) \(\approx\) \(0.776793 - 0.770802i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-2.32e3 - 3.75e3i)T \)
good2 \( 1 + (-1.51 + 1.09i)T + (39.5 - 121. i)T^{2} \)
5 \( 1 + (-76.1 + 104. i)T + (-2.41e4 - 7.43e4i)T^{2} \)
7 \( 1 + (748. + 243. i)T + (6.66e5 + 4.84e5i)T^{2} \)
13 \( 1 + (2.63e3 + 3.62e3i)T + (-1.93e7 + 5.96e7i)T^{2} \)
17 \( 1 + (1.62e4 + 1.18e4i)T + (1.26e8 + 3.90e8i)T^{2} \)
19 \( 1 + (-1.72e4 + 5.61e3i)T + (7.23e8 - 5.25e8i)T^{2} \)
23 \( 1 + 1.06e4iT - 3.40e9T^{2} \)
29 \( 1 + (-5.37e4 + 1.65e5i)T + (-1.39e10 - 1.01e10i)T^{2} \)
31 \( 1 + (-1.08e5 + 7.86e4i)T + (8.50e9 - 2.61e10i)T^{2} \)
37 \( 1 + (-1.54e5 + 4.76e5i)T + (-7.68e10 - 5.57e10i)T^{2} \)
41 \( 1 + (8.26e4 + 2.54e5i)T + (-1.57e11 + 1.14e11i)T^{2} \)
43 \( 1 + 4.58e5iT - 2.71e11T^{2} \)
47 \( 1 + (3.75e5 - 1.22e5i)T + (4.09e11 - 2.97e11i)T^{2} \)
53 \( 1 + (4.84e5 + 6.66e5i)T + (-3.63e11 + 1.11e12i)T^{2} \)
59 \( 1 + (1.72e6 + 5.61e5i)T + (2.01e12 + 1.46e12i)T^{2} \)
61 \( 1 + (-1.36e6 + 1.87e6i)T + (-9.71e11 - 2.98e12i)T^{2} \)
67 \( 1 - 3.01e6T + 6.06e12T^{2} \)
71 \( 1 + (8.76e5 - 1.20e6i)T + (-2.81e12 - 8.64e12i)T^{2} \)
73 \( 1 + (3.44e6 + 1.11e6i)T + (8.93e12 + 6.49e12i)T^{2} \)
79 \( 1 + (1.07e6 + 1.47e6i)T + (-5.93e12 + 1.82e13i)T^{2} \)
83 \( 1 + (-5.79e6 - 4.20e6i)T + (8.38e12 + 2.58e13i)T^{2} \)
89 \( 1 - 1.02e7iT - 4.42e13T^{2} \)
97 \( 1 + (5.23e6 - 3.80e6i)T + (2.49e13 - 7.68e13i)T^{2} \)
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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

−12.44991453443175313067227821899, −11.40944516519424417844046566495, −9.811099162488282413587915651035, −9.107213145720461173353402909684, −7.71873516662644580722187728140, −6.68566959512026118260801647385, −4.97644943281858332086555095536, −3.80168314174427501897074685092, −2.41982023894837619799704400506, −0.34473918371500168821619605238, 1.28620466159796104364635834998, 3.02580799798349813891182415385, 4.63958511970853588437856244660, 6.10302820151344311113395764921, 6.65882261886164063690028588984, 8.618475922499515532013368759429, 9.600276232259987333028680712634, 10.47707402285768210259502369566, 11.62484224367748663608435592038, 12.97811648467380699570077502021

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