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Properties

Label 2-2736-228.227-c0-0-7
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(1.259264460\)
\(L(\frac12)\) \(\approx\) \(1.259264460\)
\(L(1)\) 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 \( 1 - T^{2} \)
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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

−9.132873809798500117072447077502, −7.84731530472322520093311834301, −7.25760781421313360805833510494, −6.78419744042558602427863705789, −6.26070675739910094445577269080, −4.83201538347511330668355389482, −3.96457280499770947401462880832, −3.46770985732559170356566793295, −2.34550295344541057778749810199, −0.883878063637546564952788224397, 1.49038406039032023729041155104, 2.14323970983564923887728631969, 3.63931842089723274850509806091, 4.33874956176588278095412910090, 5.39359447661659859675260841311, 5.90041848213770564794817294368, 6.46984049397799711229255896215, 8.070189289378602296060400796476, 8.326642673020094850144310935679, 9.136361087223426000460471194020

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