↑

Properties

Label 2-468-117.61-c1-0-2
Degree $2$
Conductor $468$
Sign $0.857 - 0.515i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (−0.958 − 1.44i)3-s + (1.10 + 1.90i)5-s + 0.307·7-s + (−1.16 + 2.76i)9-s + (1.15 + 1.99i)11-s + (−0.675 + 3.54i)13-s + (1.69 − 3.41i)15-s + (1.19 + 2.07i)17-s + (−0.584 − 1.01i)19-s + (−0.294 − 0.443i)21-s + 5.42·23-s + (0.0716 − 0.124i)25-s + (5.10 − 0.968i)27-s + (0.871 + 1.51i)29-s + (−0.0271 − 0.0470i)31-s + ⋯
L(s)  = 1  + (−0.553 − 0.833i)3-s + (0.492 + 0.853i)5-s + 0.116·7-s + (−0.388 + 0.921i)9-s + (0.348 + 0.602i)11-s + (−0.187 + 0.982i)13-s + (0.438 − 0.882i)15-s + (0.290 + 0.502i)17-s + (−0.134 − 0.232i)19-s + (−0.0642 − 0.0968i)21-s + 1.13·23-s + (0.0143 − 0.0248i)25-s + (0.982 − 0.186i)27-s + (0.161 + 0.280i)29-s + (−0.00488 − 0.00845i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.857 - 0.515i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.857 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19937 + 0.332673i\)
\(L(\frac12)\) \(\approx\) \(1.19937 + 0.332673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.958 + 1.44i)T \)
13 \( 1 + (0.675 - 3.54i)T \)
good5 \( 1 + (-1.10 - 1.90i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.307T + 7T^{2} \)
11 \( 1 + (-1.15 - 1.99i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.19 - 2.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.584 + 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.42T + 23T^{2} \)
29 \( 1 + (-0.871 - 1.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0271 + 0.0470i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.35 - 5.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 - 5.38T + 43T^{2} \)
47 \( 1 + (0.294 - 0.510i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
59 \( 1 + (-5.48 + 9.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 4.04T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + (2.27 + 3.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.91T + 73T^{2} \)
79 \( 1 + (1.35 - 2.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 - 9.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.65 - 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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

−11.13571889320875845702124518587, −10.40964218213605965861700843650, −9.397761951700847259861907945626, −8.275814594530207128219446840735, −7.01056448532935911507942358528, −6.74865048419709389873687182049, −5.64710085666671430179012922445, −4.46985670037083697444640282236, −2.76231559658896890746824943142, −1.59551160790879627715637121616, 0.906860088874925950308382949725, 3.03187909257294074598088101209, 4.31540780365371640761375303665, 5.35658496711136753495284434136, 5.84411635599218950580939951950, 7.23975537595046957392937647611, 8.597144775722751173624798623585, 9.173717267638372787751272353888, 10.08716303953803361199246692557, 10.87318211895909580460668065767

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