Properties

Label 2-48e2-3.2-c2-0-53
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  + 1.07i·5-s + 7.21·7-s − 16.3i·11-s + 21.6·13-s − 18.9i·17-s + 17.0·19-s − 1.11i·23-s + 23.8·25-s + 29.4i·29-s − 5.63·31-s + 7.75i·35-s − 17.0·37-s − 27.4i·41-s − 52.3·43-s − 64.5i·47-s + ⋯
L(s)  = 1  + 0.214i·5-s + 1.03·7-s − 1.48i·11-s + 1.66·13-s − 1.11i·17-s + 0.897·19-s − 0.0485i·23-s + 0.953·25-s + 1.01i·29-s − 0.181·31-s + 0.221i·35-s − 0.460·37-s − 0.669i·41-s − 1.21·43-s − 1.37i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.647291443\)
\(L(\frac12)\) \(\approx\) \(2.647291443\)
\(L(2)\) 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 \( 1 - 1.07iT - 25T^{2} \)
7 \( 1 - 7.21T + 49T^{2} \)
11 \( 1 + 16.3iT - 121T^{2} \)
13 \( 1 - 21.6T + 169T^{2} \)
17 \( 1 + 18.9iT - 289T^{2} \)
19 \( 1 - 17.0T + 361T^{2} \)
23 \( 1 + 1.11iT - 529T^{2} \)
29 \( 1 - 29.4iT - 841T^{2} \)
31 \( 1 + 5.63T + 961T^{2} \)
37 \( 1 + 17.0T + 1.36e3T^{2} \)
41 \( 1 + 27.4iT - 1.68e3T^{2} \)
43 \( 1 + 52.3T + 1.84e3T^{2} \)
47 \( 1 + 64.5iT - 2.20e3T^{2} \)
53 \( 1 + 35.9iT - 2.80e3T^{2} \)
59 \( 1 - 56.8iT - 3.48e3T^{2} \)
61 \( 1 + 69.3T + 3.72e3T^{2} \)
67 \( 1 + 69.3T + 4.48e3T^{2} \)
71 \( 1 + 98.4iT - 5.04e3T^{2} \)
73 \( 1 + 37.6T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 + 7.75iT - 6.88e3T^{2} \)
89 \( 1 - 76.1iT - 7.92e3T^{2} \)
97 \( 1 - 4.84T + 9.40e3T^{2} \)
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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

−8.701668063283343104678397484997, −8.082548169524311211760928552699, −7.17944033411131598795651967950, −6.36375651558407065038204823589, −5.46926589246968852738875968566, −4.89821715316694723583477459106, −3.58447835358054931255767912615, −3.08677628830640294213885569231, −1.60734898557946565666905116229, −0.71786232759551217894640169585, 1.25041416262631540497005604027, 1.81899064377171333151863695770, 3.22672599309104584992815242368, 4.26923124679266477305336703500, 4.82574071054270874144225552558, 5.81286084893041379957916968636, 6.58623301344131398595332150032, 7.57064917518919126610922382190, 8.155308854730520738695043915378, 8.836587082878526653950811142296

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