Properties

Label 2-48e2-3.2-c2-0-55
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  + 2.44i·5-s + 3.46·7-s − 8.48i·11-s − 10.3·13-s − 12.7i·17-s + 4·19-s + 34.2i·23-s + 19·25-s − 46.5i·29-s − 38.1·31-s + 8.48i·35-s + 27.7·37-s + 29.6i·41-s − 56·43-s − 63.6i·47-s + ⋯
L(s)  = 1  + 0.489i·5-s + 0.494·7-s − 0.771i·11-s − 0.799·13-s − 0.748i·17-s + 0.210·19-s + 1.49i·23-s + 0.760·25-s − 1.60i·29-s − 1.22·31-s + 0.242i·35-s + 0.748·37-s + 0.724i·41-s − 1.30·43-s − 1.35i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.8587170536\)
\(L(\frac12)\) \(\approx\) \(0.8587170536\)
\(L(2)\) 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 \( 1 - 2.44iT - 25T^{2} \)
7 \( 1 - 3.46T + 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 + 10.3T + 169T^{2} \)
17 \( 1 + 12.7iT - 289T^{2} \)
19 \( 1 - 4T + 361T^{2} \)
23 \( 1 - 34.2iT - 529T^{2} \)
29 \( 1 + 46.5iT - 841T^{2} \)
31 \( 1 + 38.1T + 961T^{2} \)
37 \( 1 - 27.7T + 1.36e3T^{2} \)
41 \( 1 - 29.6iT - 1.68e3T^{2} \)
43 \( 1 + 56T + 1.84e3T^{2} \)
47 \( 1 + 63.6iT - 2.20e3T^{2} \)
53 \( 1 - 71.0iT - 2.80e3T^{2} \)
59 \( 1 + 101. iT - 3.48e3T^{2} \)
61 \( 1 + 69.2T + 3.72e3T^{2} \)
67 \( 1 + 28T + 4.48e3T^{2} \)
71 \( 1 - 53.8iT - 5.04e3T^{2} \)
73 \( 1 + 20T + 5.32e3T^{2} \)
79 \( 1 + 72.7T + 6.24e3T^{2} \)
83 \( 1 + 76.3iT - 6.88e3T^{2} \)
89 \( 1 - 55.1iT - 7.92e3T^{2} \)
97 \( 1 + 8T + 9.40e3T^{2} \)
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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

−8.493709765616110068156574109886, −7.68865326556192166592293777882, −7.16876557824091856160879880321, −6.19066372536253067241377633689, −5.38421577444179141327609495358, −4.63086595128369122463020202901, −3.49606872515066236519825376901, −2.73852871166277990876506453901, −1.59756060849291164150617677584, −0.20548878055316926007119508575, 1.27775596972124152608831929878, 2.20654581668163893283750854843, 3.34607045172856210343207018198, 4.60532605393144891630334899004, 4.85798600739912092335760058433, 5.91894020242106430473074646766, 6.91828719916165375505063536376, 7.50539826984479026444752851509, 8.450032575061824094393286539294, 8.954173363948047938313718938828

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