↑

Properties

Label 2-1170-13.9-c1-0-18
Degree $2$
Conductor $1170$
Sign $-0.396 + 0.918i$
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 − 5-s + (0.280 + 0.486i)7-s − 0.999·8-s + (−0.5 + 0.866i)10-s + (2.06 − 3.57i)11-s + (2.84 + 2.21i)13-s + 0.561·14-s + (−0.5 + 0.866i)16-s + (−1.56 − 2.70i)17-s + (−0.280 − 0.486i)19-s + (0.499 + 0.866i)20-s + (−2.06 − 3.57i)22-s + (2.34 − 4.05i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.106 + 0.183i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (0.621 − 1.07i)11-s + (0.788 + 0.615i)13-s + 0.150·14-s + (−0.125 + 0.216i)16-s + (−0.378 − 0.655i)17-s + (−0.0644 − 0.111i)19-s + (0.111 + 0.193i)20-s + (−0.439 − 0.761i)22-s + (0.488 − 0.845i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653137457\)
\(L(\frac12)\) \(\approx\) \(1.653137457\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (-2.84 - 2.21i)T \)
good7 \( 1 + (-0.280 - 0.486i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.06 + 3.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.56 + 2.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.280 + 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.34 + 4.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 2.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (-2.06 + 3.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.12 + 10.6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 8.56T + 53T^{2} \)
59 \( 1 + (-3.21 - 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.12 + 1.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.56 + 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.36T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 7.12T + 83T^{2} \)
89 \( 1 + (9.40 - 16.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.56 - 6.16i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.299055363939121806291124595292, −8.970154177591302216929990490305, −8.045547579788973876134005415196, −6.86256810924856419638733953742, −6.11531348065321135798161260861, −5.10652678720842929781212616922, −4.08339876811711341995963889825, −3.38033638654084112327032047813, −2.15174743755001189161073990169, −0.68842120080556210484386696999, 1.47827722220413331948394138890, 3.15931775262935576494851121271, 4.06784591873606981362631175324, 4.83485854949387857155647139973, 5.91921458071902263260351949777, 6.71807241265256384147783675249, 7.53634534990167791034305757586, 8.210007271306449211244815043923, 9.108273501167493489794262187652, 9.898281124688745594398235770556

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