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Properties

Label 2-99-9.7-c5-0-40
Degree $2$
Conductor $99$
Sign $-0.187 + 0.982i$
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  + (2.92 − 5.07i)2-s + (15.5 + 0.109i)3-s + (−1.16 − 2.01i)4-s + (−26.8 − 46.4i)5-s + (46.2 − 78.7i)6-s + (12.7 − 22.0i)7-s + 173.·8-s + (242. + 3.42i)9-s − 314.·10-s + (60.5 − 104. i)11-s + (−17.9 − 31.5i)12-s + (−376. − 652. i)13-s + (−74.6 − 129. i)14-s + (−412. − 727. i)15-s + (546. − 946. i)16-s − 352.·17-s + ⋯
L(s)  = 1  + (0.517 − 0.897i)2-s + (0.999 + 0.00703i)3-s + (−0.0364 − 0.0630i)4-s + (−0.479 − 0.830i)5-s + (0.524 − 0.893i)6-s + (0.0982 − 0.170i)7-s + 0.960·8-s + (0.999 + 0.0140i)9-s − 0.993·10-s + (0.150 − 0.261i)11-s + (−0.0359 − 0.0633i)12-s + (−0.618 − 1.07i)13-s + (−0.101 − 0.176i)14-s + (−0.473 − 0.834i)15-s + (0.533 − 0.924i)16-s − 0.295·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.187 + 0.982i$
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: \((2,\ 99,\ (\ :5/2),\ -0.187 + 0.982i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.20837 - 2.66978i\)
\(L(\frac12)\) \(\approx\) \(2.20837 - 2.66978i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.5 - 0.109i)T \)
11 \( 1 + (-60.5 + 104. i)T \)
good2 \( 1 + (-2.92 + 5.07i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (26.8 + 46.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-12.7 + 22.0i)T + (-8.40e3 - 1.45e4i)T^{2} \)
13 \( 1 + (376. + 652. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 352.T + 1.41e6T^{2} \)
19 \( 1 - 1.08e3T + 2.47e6T^{2} \)
23 \( 1 + (-914. - 1.58e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (325. - 563. i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (2.35e3 + 4.07e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.21e4T + 6.93e7T^{2} \)
41 \( 1 + (-6.16e3 - 1.06e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (976. - 1.69e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (8.17e3 - 1.41e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 133.T + 4.18e8T^{2} \)
59 \( 1 + (-1.76e4 - 3.05e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-878. + 1.52e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.03e4 - 3.52e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.20e4T + 1.80e9T^{2} \)
73 \( 1 + 1.22e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.97e4 + 6.88e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (8.49e3 - 1.47e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 5.61e4T + 5.58e9T^{2} \)
97 \( 1 + (4.30e4 - 7.45e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.74370531907116189151131239421, −11.84412045053687146507928770356, −10.61757963760908127441460577383, −9.436501176177424318614958352630, −8.177239206957239716849790193139, −7.38813409977935065813550416541, −5.00556054370435380982546452644, −3.86651551941665389190561464554, −2.75333604064519441774245996863, −1.15271891853230288927638952974, 2.02178681892149209065002236877, 3.67803546019950723375227911000, 4.98471132425892709795829161756, 6.87241552911513545739780353465, 7.19790890883566488141191609575, 8.590552972781015005166328324573, 9.891582704201202654915329440399, 11.06959323601354477446189516015, 12.44215912968122608498463108243, 13.82467276015741795157423805031

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