Properties

Label 2-2736-12.11-c1-0-1
Degree $2$
Conductor $2736$
Sign $-0.816 + 0.577i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.55i·5-s + 3.63i·7-s − 4.12·11-s − 1.20·13-s + 1.58i·17-s + i·19-s − 3.68·23-s − 7.67·25-s + 0.561i·29-s − 5.20i·31-s − 12.9·35-s + 9.27·37-s + 3.72i·41-s + 9.46i·43-s − 6.33·47-s + ⋯
L(s)  = 1  + 1.59i·5-s + 1.37i·7-s − 1.24·11-s − 0.334·13-s + 0.384i·17-s + 0.229i·19-s − 0.767·23-s − 1.53·25-s + 0.104i·29-s − 0.934i·31-s − 2.18·35-s + 1.52·37-s + 0.582i·41-s + 1.44i·43-s − 0.924·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7526392994\)
\(L(\frac12)\) \(\approx\) \(0.7526392994\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 3.55iT - 5T^{2} \)
7 \( 1 - 3.63iT - 7T^{2} \)
11 \( 1 + 4.12T + 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 - 0.561iT - 29T^{2} \)
31 \( 1 + 5.20iT - 31T^{2} \)
37 \( 1 - 9.27T + 37T^{2} \)
41 \( 1 - 3.72iT - 41T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 + 6.33T + 47T^{2} \)
53 \( 1 + 11.9iT - 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 2.60T + 61T^{2} \)
67 \( 1 - 2.06iT - 67T^{2} \)
71 \( 1 + 7.36T + 71T^{2} \)
73 \( 1 + 0.363T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 + 3.68T + 83T^{2} \)
89 \( 1 - 6.79iT - 89T^{2} \)
97 \( 1 - 8.86T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.491859958707166328534094912709, −8.219140879559653919189742132191, −7.923653249627433357945917444424, −6.95710826437214719704848629669, −6.10645745440260346818499336555, −5.69704854603611761411886991248, −4.63112473708015384322303211343, −3.40072433552387701247497601502, −2.62742486919491608503829380071, −2.13404355094132444202439920151, 0.25211806297622808022463856519, 1.15852917487542069099578525480, 2.43363409367303537087961116381, 3.74820221559638641360680531150, 4.53550599134194348142974615469, 5.07597924725074514101586725964, 5.87840919920388394485839822179, 7.09739681812882761372711961763, 7.66796765404590518413600836033, 8.334702443833910895272031484170

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