↑

Properties

Label 2-99-33.14-c2-0-2
Degree $2$
Conductor $99$
Sign $-0.578 - 0.815i$
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  + (1.75 + 2.41i)2-s + (−1.50 + 4.64i)4-s + (−4.61 + 6.35i)5-s + (0.867 − 2.67i)7-s + (−2.50 + 0.812i)8-s − 23.4·10-s + (8.89 + 6.47i)11-s + (1.22 − 0.892i)13-s + (7.95 − 2.58i)14-s + (9.45 + 6.87i)16-s + (15.5 − 21.3i)17-s + (−4.49 − 13.8i)19-s + (−22.5 − 31.0i)20-s + (−0.0177 + 32.7i)22-s + 25.0i·23-s + ⋯
L(s)  = 1  + (0.875 + 1.20i)2-s + (−0.377 + 1.16i)4-s + (−0.923 + 1.27i)5-s + (0.123 − 0.381i)7-s + (−0.312 + 0.101i)8-s − 2.34·10-s + (0.808 + 0.588i)11-s + (0.0944 − 0.0686i)13-s + (0.568 − 0.184i)14-s + (0.591 + 0.429i)16-s + (0.912 − 1.25i)17-s + (−0.236 − 0.727i)19-s + (−1.12 − 1.55i)20-s + (−0.000808 + 1.49i)22-s + 1.09i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.578 - 0.815i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.578 - 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.851232 + 1.64720i\)
\(L(\frac12)\) \(\approx\) \(0.851232 + 1.64720i\)
\(L(2)\) 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 \( 1 + (-8.89 - 6.47i)T \)
good2 \( 1 + (-1.75 - 2.41i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (4.61 - 6.35i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-0.867 + 2.67i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-1.22 + 0.892i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-15.5 + 21.3i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (4.49 + 13.8i)T + (-292. + 212. i)T^{2} \)
23 \( 1 - 25.0iT - 529T^{2} \)
29 \( 1 + (48.2 + 15.6i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-41.7 + 30.3i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (6.04 - 18.5i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-29.4 + 9.55i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 2.11T + 1.84e3T^{2} \)
47 \( 1 + (38.2 - 12.4i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (24.1 + 33.2i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (91.3 + 29.6i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-2.34 - 1.70i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 88.2T + 4.48e3T^{2} \)
71 \( 1 + (6.65 - 9.16i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-32.3 + 99.6i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (13.5 - 9.86i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-13.3 + 18.4i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 30.7iT - 7.92e3T^{2} \)
97 \( 1 + (-23.0 + 16.7i)T + (2.90e3 - 8.94e3i)T^{2} \)
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

−14.24875879735945128788570821414, −13.45298801093585167685356167907, −11.96555046361047393882935850773, −11.10611491983642500956152456541, −9.658830581150284582108038551593, −7.67379149437597496229344046168, −7.28387627822906822632880949005, −6.17000173187067485959488956382, −4.56142756059112740534386023647, −3.40555087362646243923325174143, 1.36479629201127217745175669454, 3.55209387323580776144399307825, 4.47994107917520401659287801790, 5.79382856636498112772617082624, 8.001058265269229309565375692285, 8.937486969796471802074826157162, 10.46137613609071122803342142643, 11.55666643005908225256667811393, 12.35288804668443238418152150346, 12.76171576129345252087120005165

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