Properties

Label 2-48e2-3.2-c2-0-29
Degree $2$
Conductor $2304$
Sign $0.577 + 0.816i$
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  + 4.24i·5-s − 8.48·7-s + 4i·11-s − 18·13-s − 4.24i·17-s − 16.9·19-s + 36i·23-s + 7.00·25-s − 12.7i·29-s − 8.48·31-s − 35.9i·35-s − 36·37-s − 29.6i·41-s + 67.8·43-s + 36i·47-s + ⋯
L(s)  = 1  + 0.848i·5-s − 1.21·7-s + 0.363i·11-s − 1.38·13-s − 0.249i·17-s − 0.893·19-s + 1.56i·23-s + 0.280·25-s − 0.438i·29-s − 0.273·31-s − 1.02i·35-s − 0.972·37-s − 0.724i·41-s + 1.57·43-s + 0.765i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6295468378\)
\(L(\frac12)\) \(\approx\) \(0.6295468378\)
\(L(2)\) 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 \( 1 - 4.24iT - 25T^{2} \)
7 \( 1 + 8.48T + 49T^{2} \)
11 \( 1 - 4iT - 121T^{2} \)
13 \( 1 + 18T + 169T^{2} \)
17 \( 1 + 4.24iT - 289T^{2} \)
19 \( 1 + 16.9T + 361T^{2} \)
23 \( 1 - 36iT - 529T^{2} \)
29 \( 1 + 12.7iT - 841T^{2} \)
31 \( 1 + 8.48T + 961T^{2} \)
37 \( 1 + 36T + 1.36e3T^{2} \)
41 \( 1 + 29.6iT - 1.68e3T^{2} \)
43 \( 1 - 67.8T + 1.84e3T^{2} \)
47 \( 1 - 36iT - 2.20e3T^{2} \)
53 \( 1 + 80.6iT - 2.80e3T^{2} \)
59 \( 1 - 80iT - 3.48e3T^{2} \)
61 \( 1 - 36T + 3.72e3T^{2} \)
67 \( 1 + 118.T + 4.48e3T^{2} \)
71 \( 1 + 108iT - 5.04e3T^{2} \)
73 \( 1 - 56T + 5.32e3T^{2} \)
79 \( 1 + 25.4T + 6.24e3T^{2} \)
83 \( 1 + 76iT - 6.88e3T^{2} \)
89 \( 1 + 89.0iT - 7.92e3T^{2} \)
97 \( 1 - 104T + 9.40e3T^{2} \)
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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

−8.947227701632453445549038064269, −7.59149697259141130171203206660, −7.22109864154938316093395209659, −6.47686210757362569061280899583, −5.69393668915231931226153486815, −4.68555042146248297495289192696, −3.63635990454129093624345137469, −2.88320748488929123990440337544, −2.03326848536155165884815230798, −0.21365290439948183879069952290, 0.71556219710755133327387816191, 2.22578019307554443463988936915, 3.09312896121628518259025096678, 4.21199786160443290383008243766, 4.89685206657229029684338896292, 5.84602921663563060761745103298, 6.62089137201346839929858783150, 7.30592645852562295465737875127, 8.400324698063727126682840576854, 8.891849263693138499948946349542

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