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Properties

Label 2-99-99.32-c1-0-0
Degree $2$
Conductor $99$
Sign $0.104 - 0.994i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 − 2.24i)2-s + (−1.72 + 0.120i)3-s + (−2.36 + 4.09i)4-s + (−0.137 − 0.0793i)5-s + (2.51 + 3.72i)6-s + (−2.91 + 1.68i)7-s + 7.08·8-s + (2.97 − 0.416i)9-s + 0.411i·10-s + (−3.29 + 0.349i)11-s + (3.59 − 7.36i)12-s + (−0.897 − 0.518i)13-s + (7.57 + 4.37i)14-s + (0.246 + 0.120i)15-s + (−4.45 − 7.72i)16-s − 5.02·17-s + ⋯
L(s)  = 1  + (−0.917 − 1.58i)2-s + (−0.997 + 0.0695i)3-s + (−1.18 + 2.04i)4-s + (−0.0614 − 0.0354i)5-s + (1.02 + 1.52i)6-s + (−1.10 + 0.636i)7-s + 2.50·8-s + (0.990 − 0.138i)9-s + 0.130i·10-s + (−0.994 + 0.105i)11-s + (1.03 − 2.12i)12-s + (−0.249 − 0.143i)13-s + (2.02 + 1.16i)14-s + (0.0637 + 0.0311i)15-s + (−1.11 − 1.93i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.104 - 0.994i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0351416 + 0.0316536i\)
\(L(\frac12)\) \(\approx\) \(0.0351416 + 0.0316536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.120i)T \)
11 \( 1 + (3.29 - 0.349i)T \)
good2 \( 1 + (1.29 + 2.24i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.137 + 0.0793i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.91 - 1.68i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.897 + 0.518i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 - 5.11iT - 19T^{2} \)
23 \( 1 + (4.02 + 2.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.673 - 1.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.62 + 2.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + (-0.940 + 1.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.24 + 4.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.996 - 0.575i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.36iT - 53T^{2} \)
59 \( 1 + (4.69 + 2.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.61 - 5.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.67 - 4.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.558iT - 71T^{2} \)
73 \( 1 + 6.24iT - 73T^{2} \)
79 \( 1 + (3.72 - 2.14i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.61 - 7.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.98iT - 89T^{2} \)
97 \( 1 + (-0.917 - 1.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.41511418551851919421361646072, −12.47055229481928694556972571485, −12.08027109218952447470713875906, −10.76972163438261475900969624859, −10.14916806257240917741403860048, −9.213568908593206368837740221555, −7.83220623813861440662873700722, −6.06358063221967172357902908069, −4.20760946877219518924700137708, −2.45937526355125522399995504035, 0.079226991090622365259448303123, 4.69409611559307123687617120167, 5.99620588098605386911832790040, 6.86053842002283602221456674714, 7.70200460413663060236135518593, 9.283744153975850266850286294055, 10.12410328224363895514715601526, 11.13639067674405762602948734353, 12.98294887519768085224304093771, 13.69722624348831042127186991114

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