↑

Properties

Label 2-2736-19.7-c1-0-8
Degree $2$
Conductor $2736$
Sign $0.988 - 0.150i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + (−1.87 − 3.25i)5-s − 4.75·7-s − 4.36·11-s + (−0.5 + 0.866i)13-s + (3.06 + 5.30i)17-s + (−0.694 + 4.30i)19-s + (1.87 − 3.25i)23-s + (−4.56 + 7.90i)25-s + (2.69 − 4.66i)29-s − 7.36·31-s + (8.94 + 15.4i)35-s − 7.12·37-s + (−1.75 − 3.04i)41-s + (−0.379 − 0.657i)43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (−0.840 − 1.45i)5-s − 1.79·7-s − 1.31·11-s + (−0.138 + 0.240i)13-s + (0.743 + 1.28i)17-s + (−0.159 + 0.987i)19-s + (0.391 − 0.678i)23-s + (−0.912 + 1.58i)25-s + (0.500 − 0.866i)29-s − 1.32·31-s + (1.51 + 2.61i)35-s − 1.17·37-s + (−0.274 − 0.475i)41-s + (−0.0578 − 0.100i)43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.988 - 0.150i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5351799569\)
\(L(\frac12)\) \(\approx\) \(0.5351799569\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (0.694 - 4.30i)T \)
good5 \( 1 + (1.87 + 3.25i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.75T + 7T^{2} \)
11 \( 1 + 4.36T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.06 - 5.30i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.87 + 3.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.69 + 4.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + (1.75 + 3.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.379 + 0.657i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.57 + 4.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.63 + 8.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.25 + 7.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.684 + 1.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.82 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.25 + 3.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.07 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.90T + 83T^{2} \)
89 \( 1 + (-4.18 + 7.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.82 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.689446541377276274897102723496, −8.225082450504405515132595096093, −7.47085409569188840816303702169, −6.52002016920086608631157833727, −5.65040888339114859288395710785, −5.03261207237728666345258911448, −3.83935234262277453828839534167, −3.50187922469199574540513500570, −2.10468865880935906014590315756, −0.57081013864697300519308865122, 0.31961450550133631124000580076, 2.68369978257664019530599716032, 3.02098460082226700749152060846, 3.62563884768847078678941583624, 4.97159578832318405794986725504, 5.82560183618334882953392935082, 6.79171646164033702523863517305, 7.23748322544292843021949554937, 7.68019548499577434758997809420, 8.950397819645861778102846541002

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