Properties

Label 2-48e2-4.3-c2-0-28
Degree $2$
Conductor $2304$
Sign $-i$
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  − 4.89·5-s + 7.74i·7-s + 12.6i·11-s + 15.4·13-s + 25.2·17-s + 8i·19-s − 39.1i·23-s − 1.00·25-s + 24.4·29-s + 7.74i·31-s − 37.9i·35-s + 46.4·37-s − 25.2·41-s + 40i·43-s − 39.1i·47-s + ⋯
L(s)  = 1  − 0.979·5-s + 1.10i·7-s + 1.14i·11-s + 1.19·13-s + 1.48·17-s + 0.421i·19-s − 1.70i·23-s − 0.0400·25-s + 0.844·29-s + 0.249i·31-s − 1.08i·35-s + 1.25·37-s − 0.617·41-s + 0.930i·43-s − 0.833i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.723913073\)
\(L(\frac12)\) \(\approx\) \(1.723913073\)
\(L(2)\) 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 \( 1 + 4.89T + 25T^{2} \)
7 \( 1 - 7.74iT - 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 - 15.4T + 169T^{2} \)
17 \( 1 - 25.2T + 289T^{2} \)
19 \( 1 - 8iT - 361T^{2} \)
23 \( 1 + 39.1iT - 529T^{2} \)
29 \( 1 - 24.4T + 841T^{2} \)
31 \( 1 - 7.74iT - 961T^{2} \)
37 \( 1 - 46.4T + 1.36e3T^{2} \)
41 \( 1 + 25.2T + 1.68e3T^{2} \)
43 \( 1 - 40iT - 1.84e3T^{2} \)
47 \( 1 + 39.1iT - 2.20e3T^{2} \)
53 \( 1 + 14.6T + 2.80e3T^{2} \)
59 \( 1 + 25.2iT - 3.48e3T^{2} \)
61 \( 1 + 15.4T + 3.72e3T^{2} \)
67 \( 1 - 80iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 10T + 5.32e3T^{2} \)
79 \( 1 + 54.2iT - 6.24e3T^{2} \)
83 \( 1 - 139. iT - 6.88e3T^{2} \)
89 \( 1 - 50.5T + 7.92e3T^{2} \)
97 \( 1 - 50T + 9.40e3T^{2} \)
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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

−8.834406251397733836800315191977, −8.220423723283660770334198584471, −7.69180178646834531093593637916, −6.63599228865862017144903352118, −5.95064394926952496565319672353, −4.98940452858962654112110639690, −4.17133987119531246443610483720, −3.29032718534897641218692282409, −2.29774717646677741164165326037, −1.02746633723292906860017259548, 0.55282066410150342486339391362, 1.30159299574085741717607401325, 3.24115640023573589944094193318, 3.57235601232480537145856469239, 4.41097937302263102663450355029, 5.57138875852656833396716126078, 6.25875036884381451700510437452, 7.34882993034128644065902036848, 7.80998320382724542304806654685, 8.435495144994788073696155631406

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