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Properties

Label 2-48e2-24.11-c1-0-30
Degree $2$
Conductor $2304$
Sign $-0.985 + 0.169i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.585·5-s − 0.828i·7-s − 2.82i·11-s − 2.82i·13-s − 2.58i·17-s − 5.65·19-s − 6.82·23-s − 4.65·25-s − 3.41·29-s + 8.82i·31-s − 0.485i·35-s + 7.65i·37-s + 5.41i·41-s + 1.65·43-s − 4.48·47-s + ⋯
L(s)  = 1  + 0.261·5-s − 0.313i·7-s − 0.852i·11-s − 0.784i·13-s − 0.627i·17-s − 1.29·19-s − 1.42·23-s − 0.931·25-s − 0.634·29-s + 1.58i·31-s − 0.0820i·35-s + 1.25i·37-s + 0.845i·41-s + 0.252·43-s − 0.654·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4647990727\)
\(L(\frac12)\) \(\approx\) \(0.4647990727\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 5T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 - 8.82iT - 31T^{2} \)
37 \( 1 - 7.65iT - 37T^{2} \)
41 \( 1 - 5.41iT - 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 + 13.6iT - 59T^{2} \)
61 \( 1 + 3.65iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 - 3.75iT - 89T^{2} \)
97 \( 1 - 2.34T + 97T^{2} \)
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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

−8.469876201794516617174647470508, −8.059508385014652739549220968848, −7.06976349748854356152661916998, −6.21334430589839571823284236589, −5.60954251541041735328909275286, −4.61679690207275194865865715882, −3.68172223563678435722019297893, −2.78637633829103657988950457794, −1.60036584838745865650294246436, −0.14521879525570596710742023155, 1.88315417137459952637792387924, 2.31309741102351129722389550244, 4.10046254065773747266944469205, 4.19161724723050316145218186929, 5.74321423432464136038343047358, 6.02824321065049013239422844552, 7.13723836688230777107165993107, 7.76175165049004360516421265597, 8.735089165159183120672806201126, 9.288809027281021738099627623361

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