↑

Properties

Label 2-2736-76.27-c1-0-3
Degree $2$
Conductor $2736$
Sign $-0.952 - 0.305i$
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  + (−0.5 + 0.866i)5-s + 2i·7-s − 2i·11-s + (−1.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (−1.73 + 4i)19-s + (−0.866 + 0.5i)23-s + (2 + 3.46i)25-s + (1.5 − 0.866i)29-s + 3.46·31-s + (−1.73 − i)35-s − 3.46i·37-s + (−4.5 − 2.59i)41-s + (−7.79 − 4.5i)43-s + (−9.52 + 5.5i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + 0.755i·7-s − 0.603i·11-s + (−0.416 + 0.240i)13-s + (0.121 − 0.210i)17-s + (−0.397 + 0.917i)19-s + (−0.180 + 0.104i)23-s + (0.400 + 0.692i)25-s + (0.278 − 0.160i)29-s + 0.622·31-s + (−0.292 − 0.169i)35-s − 0.569i·37-s + (−0.702 − 0.405i)41-s + (−1.18 − 0.686i)43-s + (−1.38 + 0.802i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6523563583\)
\(L(\frac12)\) \(\approx\) \(0.6523563583\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 4i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 3.46iT - 37T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.79 + 4.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.52 - 5.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 + 4.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.79 - 13.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.5 + 7.79i)T + (48.5 + 84.0i)T^{2} \)
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

−9.047846130555781966424975127002, −8.456782755506304499740576026125, −7.69073648759321620002989307070, −6.87748453472931815459277597935, −6.06939983653535017251783937446, −5.40511527171245735492093291392, −4.44920260702265117788682763949, −3.42838165925662739545906959285, −2.68358192850543121006834349244, −1.54582567321897168141908514444, 0.20801933040428733529516496316, 1.49580706300735824716269111324, 2.71162587298956414453872142111, 3.71695016613478938663677489014, 4.73449405506465072432651548618, 4.99823712262728120447758172431, 6.52467375436696188650127633915, 6.77270229545537153711335204065, 7.943106763371559682908126652658, 8.250784459535147274206399532820

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