Properties

Label 2-48e2-24.5-c2-0-44
Degree $2$
Conductor $2304$
Sign $0.985 - 0.169i$
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  + 8.04·5-s + 2·7-s + 21.7·11-s + 17.3i·13-s − 11.8i·17-s + 10.7i·19-s − 35.0i·23-s + 39.7·25-s + 11.7·29-s + 35.5·31-s + 16.0·35-s − 26i·37-s − 2.28i·41-s + 65.5i·43-s + 27.2i·47-s + ⋯
L(s)  = 1  + 1.60·5-s + 0.285·7-s + 1.97·11-s + 1.33i·13-s − 0.697i·17-s + 0.566i·19-s − 1.52i·23-s + 1.59·25-s + 0.405·29-s + 1.14·31-s + 0.459·35-s − 0.702i·37-s − 0.0558i·41-s + 1.52i·43-s + 0.578i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.985 - 0.169i$
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: \((2,\ 2304,\ (\ :1),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.687713212\)
\(L(\frac12)\) \(\approx\) \(3.687713212\)
\(L(2)\) 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 \( 1 - 8.04T + 25T^{2} \)
7 \( 1 - 2T + 49T^{2} \)
11 \( 1 - 21.7T + 121T^{2} \)
13 \( 1 - 17.3iT - 169T^{2} \)
17 \( 1 + 11.8iT - 289T^{2} \)
19 \( 1 - 10.7iT - 361T^{2} \)
23 \( 1 + 35.0iT - 529T^{2} \)
29 \( 1 - 11.7T + 841T^{2} \)
31 \( 1 - 35.5T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 + 2.28iT - 1.68e3T^{2} \)
43 \( 1 - 65.5iT - 1.84e3T^{2} \)
47 \( 1 - 27.2iT - 2.20e3T^{2} \)
53 \( 1 + 49.5T + 2.80e3T^{2} \)
59 \( 1 - 73.5T + 3.48e3T^{2} \)
61 \( 1 + 7.52iT - 3.72e3T^{2} \)
67 \( 1 + 65.2iT - 4.48e3T^{2} \)
71 \( 1 - 84.1iT - 5.04e3T^{2} \)
73 \( 1 + 84.5T + 5.32e3T^{2} \)
79 \( 1 + 75.5T + 6.24e3T^{2} \)
83 \( 1 - 48.2T + 6.88e3T^{2} \)
89 \( 1 + 146. iT - 7.92e3T^{2} \)
97 \( 1 + 106.T + 9.40e3T^{2} \)
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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

−9.042052304538677028938275017334, −8.316940847548168915678422014063, −6.94177513251909241022049800427, −6.46102904855060479354400301774, −5.97645026284259884770795431019, −4.75006675876614938301144943380, −4.20655370559790048504870701149, −2.82959105943339555920907841330, −1.83691579104503388504819405497, −1.16446102512712198324041686475, 1.08813813563561251718466576604, 1.73634088311007911152189066516, 2.91123295143886446140082856655, 3.88112308473603589953883490640, 5.01984717006894931152566570336, 5.73958348745143542610824922772, 6.37548884503880358335358836806, 7.05005715401751213020124833118, 8.229360565465715271161887962748, 8.879440227070539132199777499939

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