↑

Properties

Label 2-99-11.2-c2-0-6
Degree $2$
Conductor $99$
Sign $0.812 - 0.582i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.47 + 1.12i)2-s + (7.54 + 5.48i)4-s + (−1.69 − 5.20i)5-s + (−4.20 + 5.79i)7-s + (11.4 + 15.7i)8-s − 19.9i·10-s + (−0.170 − 10.9i)11-s + (−6.27 − 2.03i)13-s + (−21.1 + 15.3i)14-s + (10.4 + 32.0i)16-s + (−17.5 + 5.70i)17-s + (4.95 + 6.81i)19-s + (15.7 − 48.5i)20-s + (11.8 − 38.3i)22-s + 17.7·23-s + ⋯
L(s)  = 1  + (1.73 + 0.564i)2-s + (1.88 + 1.37i)4-s + (−0.338 − 1.04i)5-s + (−0.601 + 0.827i)7-s + (1.43 + 1.96i)8-s − 1.99i·10-s + (−0.0155 − 0.999i)11-s + (−0.482 − 0.156i)13-s + (−1.51 + 1.09i)14-s + (0.651 + 2.00i)16-s + (−1.03 + 0.335i)17-s + (0.260 + 0.358i)19-s + (0.788 − 2.42i)20-s + (0.537 − 1.74i)22-s + 0.770·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.812 - 0.582i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.80912 + 0.902746i\)
\(L(\frac12)\) \(\approx\) \(2.80912 + 0.902746i\)
\(L(2)\) 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 \( 1 + (0.170 + 10.9i)T \)
good2 \( 1 + (-3.47 - 1.12i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (1.69 + 5.20i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (4.20 - 5.79i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.27 + 2.03i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (17.5 - 5.70i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-4.95 - 6.81i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 17.7T + 529T^{2} \)
29 \( 1 + (11.8 - 16.2i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-10.9 + 33.6i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-39.4 - 28.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (18.5 + 25.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 45.0iT - 1.84e3T^{2} \)
47 \( 1 + (0.589 - 0.428i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (21.6 - 66.7i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-19.5 - 14.1i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (39.0 - 12.6i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 96.0T + 4.48e3T^{2} \)
71 \( 1 + (12.2 + 37.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-41.2 + 56.8i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (84.9 + 27.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (24.9 - 8.10i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 118.T + 7.92e3T^{2} \)
97 \( 1 + (10.2 - 31.4i)T + (-7.61e3 - 5.53e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.54525517354596454030979854345, −12.87927752339058631000676737025, −12.15214640991355332946331504694, −11.18136164462083387633457063527, −9.116334423534923078151988937553, −7.957809144390289612928939765345, −6.43202575165707534719086040760, −5.48374903982273095978843774835, −4.37664881107825448282531031230, −2.91339274456622739971943642138, 2.52972764831246346752417644940, 3.76366000992969406301215453479, 4.91118116688188889543023467133, 6.69939960678341097431938102285, 7.12535134986903930760885422649, 9.790210929084419835146974150041, 10.78613018402976019109224169690, 11.54514061669953020343829128640, 12.70748971448722719038654241499, 13.47369319648044443030597475747

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