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Properties

Label 2-48e2-144.43-c0-0-2
Degree $2$
Conductor $2304$
Sign $0.953 + 0.300i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (0.965 − 0.258i)3-s + (−0.366 + 1.36i)5-s + (0.707 − 1.22i)7-s + (0.866 − 0.499i)9-s + (−0.258 − 0.965i)11-s + 1.41i·15-s + 17-s + (−0.707 − 0.707i)19-s + (0.366 − 1.36i)21-s + (−0.866 − 0.5i)25-s + (0.707 − 0.707i)27-s + (0.366 + 1.36i)29-s + (−0.499 − 0.866i)33-s + (1.41 + 1.41i)35-s + (−1 − i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + (−0.366 + 1.36i)5-s + (0.707 − 1.22i)7-s + (0.866 − 0.499i)9-s + (−0.258 − 0.965i)11-s + 1.41i·15-s + 17-s + (−0.707 − 0.707i)19-s + (0.366 − 1.36i)21-s + (−0.866 − 0.5i)25-s + (0.707 − 0.707i)27-s + (0.366 + 1.36i)29-s + (−0.499 − 0.866i)33-s + (1.41 + 1.41i)35-s + (−1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703811804\)
\(L(\frac12)\) \(\approx\) \(1.703811804\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
good5 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.972064740696005232936244183584, −8.223170154867263170447115235859, −7.58154223101178459296490924668, −7.05284743308028303919882273058, −6.40162928802839022238359872093, −5.07728560374179092306209229837, −3.95079020511146600356510278953, −3.37398346314212910867327131235, −2.60374161004855603992899089588, −1.24116312255139464320474665022, 1.59259282242201618923593354535, 2.29412991115257954071080001548, 3.57289834815688016145081627753, 4.54781502833084385001303727447, 5.01940228586431255168620206664, 5.86491329251523034845417418985, 7.24192668277106700378019519581, 8.073840884031902788098193355898, 8.444028492019798046690935179143, 8.983353162861912947732428027592

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