Properties

Label 2-2736-228.227-c0-0-1
Degree $2$
Conductor $2736$
Sign $-0.816 - 0.577i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s + 2i·7-s − 1.41·11-s − 1.41i·17-s + i·19-s + 1.41·23-s − 1.00·25-s − 2.82·35-s − 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s − 2.82i·77-s − 1.41·83-s + 2.00·85-s + ⋯
L(s)  = 1  + 1.41i·5-s + 2i·7-s − 1.41·11-s − 1.41i·17-s + i·19-s + 1.41·23-s − 1.00·25-s − 2.82·35-s − 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s − 2.82i·77-s − 1.41·83-s + 2.00·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9469997819\)
\(L(\frac12)\) \(\approx\) \(0.9469997819\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - 2iT - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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

−9.400173048581408354543022018356, −8.496132847076799336790071366200, −7.81849461862177045661346444372, −7.00231306528612849716364554609, −6.25217281058969995526214465318, −5.42344790612775697933073101832, −4.96264673909214643984263046458, −3.23332636057064379034778511041, −2.81507980698047955357329210594, −2.12337318778334656757748351126, 0.59956485099503940508533282192, 1.58693355816893826856191716342, 3.11381557447035940134530015061, 4.11610068772940254664088502274, 4.76650149561649759856095958143, 5.32054269633031846868438154301, 6.54478891107410618145235976902, 7.29871827647541891227300543365, 8.032131492031980649612570800033, 8.525205382485707880562984109575

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