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Properties

Label 2-48e2-24.5-c2-0-21
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  + 5.21·5-s − 2·7-s − 4.78·11-s − 1.38i·13-s − 14.6i·17-s + 26.7i·19-s + 18.0i·23-s + 2.23·25-s − 25.0·29-s + 39.5·31-s − 10.4·35-s − 26i·37-s + 28.8i·41-s + 9.52i·43-s + 80.2i·47-s + ⋯
L(s)  = 1  + 1.04·5-s − 0.285·7-s − 0.434·11-s − 0.106i·13-s − 0.863i·17-s + 1.40i·19-s + 0.784i·23-s + 0.0895·25-s − 0.862·29-s + 1.27·31-s − 0.298·35-s − 0.702i·37-s + 0.702i·41-s + 0.221i·43-s + 1.70i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.842421947\)
\(L(\frac12)\) \(\approx\) \(1.842421947\)
\(L(2)\) 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 \( 1 - 5.21T + 25T^{2} \)
7 \( 1 + 2T + 49T^{2} \)
11 \( 1 + 4.78T + 121T^{2} \)
13 \( 1 + 1.38iT - 169T^{2} \)
17 \( 1 + 14.6iT - 289T^{2} \)
19 \( 1 - 26.7iT - 361T^{2} \)
23 \( 1 - 18.0iT - 529T^{2} \)
29 \( 1 + 25.0T + 841T^{2} \)
31 \( 1 - 39.5T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 - 9.52iT - 1.84e3T^{2} \)
47 \( 1 - 80.2iT - 2.20e3T^{2} \)
53 \( 1 - 9.79T + 2.80e3T^{2} \)
59 \( 1 - 73.5T + 3.48e3T^{2} \)
61 \( 1 - 67.5iT - 3.72e3T^{2} \)
67 \( 1 - 102. iT - 4.48e3T^{2} \)
71 \( 1 + 21.9iT - 5.04e3T^{2} \)
73 \( 1 - 140.T + 5.32e3T^{2} \)
79 \( 1 - 0.476T + 6.24e3T^{2} \)
83 \( 1 + 31.3T + 6.88e3T^{2} \)
89 \( 1 + 13.1iT - 7.92e3T^{2} \)
97 \( 1 + 69.2T + 9.40e3T^{2} \)
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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

−9.210934782235618485353891376425, −8.156166021134943123489563982882, −7.52619164912787014300578656436, −6.53977509061774630416316273141, −5.80509222653689646323124623052, −5.27292008925437557247597825367, −4.15503360724804933122005609263, −3.09460635076697486184917344195, −2.21040701202566463950922909527, −1.15224015869606057604241892388, 0.44964930339365080587022211099, 1.87033149664828290444242965585, 2.60681576753113305247673364300, 3.70331697178029426298852083057, 4.81613357831769654344225205857, 5.49024657412978838201520156712, 6.42033226315300414803979081844, 6.84774500387068115066557311456, 8.018894595417378847307167965710, 8.687376066409815142030395479054

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