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Properties

Label 2-99-9.4-c3-0-14
Degree $2$
Conductor $99$
Sign $0.766 - 0.642i$
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 + 3.46i)2-s + (−4.5 − 2.59i)3-s + (−3.99 + 6.92i)4-s + (9.5 − 16.4i)5-s − 20.7i·6-s + (13 + 22.5i)7-s + (13.5 + 23.3i)9-s + 76·10-s + (−5.5 − 9.52i)11-s + (35.9 − 20.7i)12-s + (28 − 48.4i)13-s + (−51.9 + 90.0i)14-s + (−85.5 + 49.3i)15-s + (31.9 + 55.4i)16-s + 104·17-s + (−54 + 93.5i)18-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.866 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.849 − 1.47i)5-s − 1.41i·6-s + (0.701 + 1.21i)7-s + (0.5 + 0.866i)9-s + 2.40·10-s + (−0.150 − 0.261i)11-s + (0.866 − 0.499i)12-s + (0.597 − 1.03i)13-s + (−0.992 + 1.71i)14-s + (−1.47 + 0.849i)15-s + (0.499 + 0.866i)16-s + 1.48·17-s + (−0.707 + 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.02503 + 0.737050i\)
\(L(\frac12)\) \(\approx\) \(2.02503 + 0.737050i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 + 2.59i)T \)
11 \( 1 + (5.5 + 9.52i)T \)
good2 \( 1 + (-2 - 3.46i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-9.5 + 16.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-13 - 22.5i)T + (-171.5 + 297. i)T^{2} \)
13 \( 1 + (-28 + 48.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 104T + 4.91e3T^{2} \)
19 \( 1 + 96T + 6.85e3T^{2} \)
23 \( 1 + (20 - 34.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (9 + 15.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (24.5 - 42.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 75T + 5.06e4T^{2} \)
41 \( 1 + (148 - 256. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (186 + 322. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-74.5 - 129. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 417T + 1.48e5T^{2} \)
59 \( 1 + (-8.5 + 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (45 + 77.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (536.5 - 929. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 285T + 3.57e5T^{2} \)
73 \( 1 + 962T + 3.89e5T^{2} \)
79 \( 1 + (298 + 516. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-249 - 431. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.23e3T + 7.04e5T^{2} \)
97 \( 1 + (-165.5 - 286. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.35290325180945714767465807494, −12.80464542472788248146474506839, −11.91208659382703445810455152534, −10.33160123042702676221379895738, −8.617662836404017644973075792835, −7.892794699095979700461419634609, −6.01812740149247243973060479215, −5.59884701762763185856090111165, −4.82308944001818247790891713683, −1.48577782499096667261737069586, 1.66138181867850470596890048767, 3.49245957150588644782494959014, 4.58452473766923163743296530520, 6.13426134563951743167373437633, 7.31552770881455236666923830531, 9.821741490660431831725402956443, 10.54621984541374083580788591065, 10.98102824510126362425107356292, 11.93681555220801856560452762363, 13.27482313549325367310274023233

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