Properties

Label 2-1170-65.4-c1-0-0
Degree $2$
Conductor $1170$
Sign $-0.992 + 0.122i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.571 − 2.16i)5-s + (0.603 + 1.04i)7-s + 0.999·8-s + (2.15 + 0.585i)10-s + (−4.46 − 2.57i)11-s + (2.24 − 2.82i)13-s − 1.20·14-s + (−0.5 + 0.866i)16-s + (−4.10 + 2.36i)17-s + (−1.84 + 1.06i)19-s + (−1.58 + 1.57i)20-s + (4.46 − 2.57i)22-s + (1.88 + 1.08i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.255 − 0.966i)5-s + (0.227 + 0.394i)7-s + 0.353·8-s + (0.682 + 0.185i)10-s + (−1.34 − 0.776i)11-s + (0.622 − 0.782i)13-s − 0.322·14-s + (−0.125 + 0.216i)16-s + (−0.994 + 0.574i)17-s + (−0.423 + 0.244i)19-s + (−0.354 + 0.352i)20-s + (0.951 − 0.549i)22-s + (0.392 + 0.226i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.002882025889\)
\(L(\frac12)\) \(\approx\) \(0.002882025889\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.571 + 2.16i)T \)
13 \( 1 + (-2.24 + 2.82i)T \)
good7 \( 1 + (-0.603 - 1.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.46 + 2.57i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.10 - 2.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.84 - 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.88 - 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 - 4.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.91iT - 31T^{2} \)
37 \( 1 + (2.20 - 3.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.10 + 2.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.70 + 0.986i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.852T + 47T^{2} \)
53 \( 1 - 4.48iT - 53T^{2} \)
59 \( 1 + (1.68 - 0.970i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.53 + 2.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.02 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.298 + 0.172i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (14.1 + 8.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.24 + 7.34i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.26795190135486526030711682172, −8.888594894451070112717682271384, −8.606321959019546989558397165053, −7.999788693004321833563804908576, −7.00987669340838800450458919456, −5.76695044228010445393054539475, −5.38317970082143946178569117736, −4.38286681403226424581741556024, −3.09294379047329820791689446443, −1.52773049885605353677150934004, 0.00136425042594675127161682147, 2.01346154133665139958262465012, 2.77594849123679850352361154004, 4.02370654188288838484395124423, 4.73211251090412357925615893556, 6.13106945256818775082057556838, 7.12943786476857914378902355879, 7.63351403793453755542243773975, 8.593596491771410750862929993101, 9.532035885997212358308082744216

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