Properties

Label 2-48e2-24.5-c2-0-24
Degree $2$
Conductor $2304$
Sign $0.169 - 0.985i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s + 8·7-s + 11.3·11-s + 8i·13-s − 12.7i·17-s + 32i·19-s + 33.9i·23-s − 23·25-s + 43.8·29-s − 40·31-s − 11.3·35-s − 26i·37-s + 66.4i·41-s − 16i·43-s + 11.3i·47-s + ⋯
L(s)  = 1  − 0.282·5-s + 1.14·7-s + 1.02·11-s + 0.615i·13-s − 0.748i·17-s + 1.68i·19-s + 1.47i·23-s − 0.920·25-s + 1.51·29-s − 1.29·31-s − 0.323·35-s − 0.702i·37-s + 1.62i·41-s − 0.372i·43-s + 0.240i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.069119050\)
\(L(\frac12)\) \(\approx\) \(2.069119050\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + 1.41T + 25T^{2} \)
7 \( 1 - 8T + 49T^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 - 8iT - 169T^{2} \)
17 \( 1 + 12.7iT - 289T^{2} \)
19 \( 1 - 32iT - 361T^{2} \)
23 \( 1 - 33.9iT - 529T^{2} \)
29 \( 1 - 43.8T + 841T^{2} \)
31 \( 1 + 40T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 - 66.4iT - 1.68e3T^{2} \)
43 \( 1 + 16iT - 1.84e3T^{2} \)
47 \( 1 - 11.3iT - 2.20e3T^{2} \)
53 \( 1 + 32.5T + 2.80e3T^{2} \)
59 \( 1 - 22.6T + 3.48e3T^{2} \)
61 \( 1 - 54iT - 3.72e3T^{2} \)
67 \( 1 + 80iT - 4.48e3T^{2} \)
71 \( 1 + 79.1iT - 5.04e3T^{2} \)
73 \( 1 + 96T + 5.32e3T^{2} \)
79 \( 1 - 104T + 6.24e3T^{2} \)
83 \( 1 - 101.T + 6.88e3T^{2} \)
89 \( 1 - 77.7iT - 7.92e3T^{2} \)
97 \( 1 + 80T + 9.40e3T^{2} \)
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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

−9.096571244655334849589673830858, −7.990776892736681279825906435449, −7.71588489447979623394327014560, −6.69409034566485266241086782579, −5.84999607375970083617114219141, −4.97470192043978893374893742831, −4.13201178127763089585665144340, −3.42829475589692927911913334673, −1.92919329826591436508112531798, −1.25945247615183311690766855539, 0.52731197680205007433539628504, 1.65981445682039772320372174541, 2.72369529685937526103488588961, 3.92405953509804139886141578797, 4.59114699715124120187282819571, 5.38736126733883013672116131371, 6.43600645853625007180072825804, 7.07754222858319564937970894983, 8.055746530747326860438290561725, 8.551569148911007970074900405631

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