Properties

Label 2-2448-51.50-c0-0-3
Degree $2$
Conductor $2448$
Sign $-0.577 + 0.816i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  − 5-s − 1.41i·7-s + 11-s − 13-s − 17-s + 19-s − 23-s − 1.41i·31-s + 1.41i·35-s − 1.41i·37-s − 41-s − 43-s − 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯
L(s)  = 1  − 5-s − 1.41i·7-s + 11-s − 13-s − 17-s + 19-s − 23-s − 1.41i·31-s + 1.41i·35-s − 1.41i·37-s − 41-s − 43-s − 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6358834322\)
\(L(\frac12)\) \(\approx\) \(0.6358834322\)
\(L(1)\) 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 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
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

−8.849846826846949466904030159627, −7.86879314197000157011998814481, −7.34676028028109217740493551294, −6.85432112542552520359496177537, −5.78369191088288333643128117643, −4.51789993306017833989556489420, −4.11710110675862527828872684700, −3.36437854922545179748816957716, −1.91616987306121324036095325205, −0.41591831571272995581069633124, 1.72376226840923333116863171027, 2.82894983947271685607974807952, 3.69633514139184534115929782871, 4.71106247976782240738257836779, 5.35158155491158432899602951978, 6.47296484370388385668093279473, 6.98176547860601372823561444636, 8.130300908233192138743690179919, 8.443662066419726348029924415745, 9.430764566231257938017958051063

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