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Properties

Label 2-99-9.7-c5-0-31
Degree $2$
Conductor $99$
Sign $0.782 + 0.622i$
Analytic cond. $15.8779$
Root an. cond. $3.98472$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.34 − 5.78i)2-s + (13.3 − 7.96i)3-s + (−6.32 − 10.9i)4-s + (39.9 + 69.2i)5-s + (−1.35 − 104. i)6-s + (−78.3 + 135. i)7-s + 129.·8-s + (115. − 213. i)9-s + 534.·10-s + (60.5 − 104. i)11-s + (−172. − 96.4i)12-s + (220. + 381. i)13-s + (523. + 907. i)14-s + (1.08e3 + 609. i)15-s + (634. − 1.09e3i)16-s + 2.04e3·17-s + ⋯
L(s)  = 1  + (0.590 − 1.02i)2-s + (0.859 − 0.511i)3-s + (−0.197 − 0.342i)4-s + (0.715 + 1.23i)5-s + (−0.0153 − 1.18i)6-s + (−0.604 + 1.04i)7-s + 0.714·8-s + (0.477 − 0.878i)9-s + 1.69·10-s + (0.150 − 0.261i)11-s + (−0.345 − 0.193i)12-s + (0.361 + 0.626i)13-s + (0.714 + 1.23i)14-s + (1.24 + 0.699i)15-s + (0.619 − 1.07i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(15.8779\)
Root analytic conductor: \(3.98472\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :5/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.67638 - 1.28411i\)
\(L(\frac12)\) \(\approx\) \(3.67638 - 1.28411i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.3 + 7.96i)T \)
11 \( 1 + (-60.5 + 104. i)T \)
good2 \( 1 + (-3.34 + 5.78i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-39.9 - 69.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (78.3 - 135. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
13 \( 1 + (-220. - 381. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 2.04e3T + 1.41e6T^{2} \)
19 \( 1 + 2.36e3T + 2.47e6T^{2} \)
23 \( 1 + (981. + 1.70e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (370. - 641. i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-402. - 697. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.17e4T + 6.93e7T^{2} \)
41 \( 1 + (3.00e3 + 5.20e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-1.11e4 + 1.92e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (5.66e3 - 9.81e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.84e3T + 4.18e8T^{2} \)
59 \( 1 + (3.72e3 + 6.45e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.84e3 - 3.19e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.42e4 + 2.46e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.79e4T + 1.80e9T^{2} \)
73 \( 1 - 8.76e4T + 2.07e9T^{2} \)
79 \( 1 + (4.94e4 - 8.55e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (8.80e3 - 1.52e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 9.25e4T + 5.58e9T^{2} \)
97 \( 1 + (-1.31e4 + 2.27e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.66083683882470974658524606019, −12.12445855005484297605647467269, −10.72537977264211361425532325597, −9.833449234145622710642738261791, −8.569762868831093551073997744677, −7.02282103544891796094880585036, −5.96552219502638751493225142138, −3.67956571746077786906674845032, −2.74312417563641935791963565813, −1.87724803856616915774518001848, 1.40398490324893255292400926021, 3.74005081299912542494712243743, 4.85300311109759975986892917853, 5.98035038334679176191636774035, 7.46298215511106888174926776322, 8.445087407759392529303617925440, 9.782077290286604776504953474802, 10.43569531153317679581982137121, 12.72930067708337820814032023663, 13.33284046017361574627095924397

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