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Properties

Label 2-1152-48.11-c3-0-33
Degree $2$
Conductor $1152$
Sign $0.888 + 0.458i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.2 + 15.2i)5-s + 24.4·7-s + (20.1 + 20.1i)11-s + (26.7 − 26.7i)13-s − 85.9i·17-s + (−53.2 − 53.2i)19-s + 119. i·23-s − 337. i·25-s + (−78.5 − 78.5i)29-s − 200. i·31-s + (−372. + 372. i)35-s + (76.9 + 76.9i)37-s − 279.·41-s + (−15.7 + 15.7i)43-s + 470.·47-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)5-s + 1.32·7-s + (0.553 + 0.553i)11-s + (0.571 − 0.571i)13-s − 1.22i·17-s + (−0.643 − 0.643i)19-s + 1.08i·23-s − 2.69i·25-s + (−0.502 − 0.502i)29-s − 1.16i·31-s + (−1.79 + 1.79i)35-s + (0.342 + 0.342i)37-s − 1.06·41-s + (−0.0559 + 0.0559i)43-s + 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.609855720\)
\(L(\frac12)\) \(\approx\) \(1.609855720\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.2 - 15.2i)T - 125iT^{2} \)
7 \( 1 - 24.4T + 343T^{2} \)
11 \( 1 + (-20.1 - 20.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-26.7 + 26.7i)T - 2.19e3iT^{2} \)
17 \( 1 + 85.9iT - 4.91e3T^{2} \)
19 \( 1 + (53.2 + 53.2i)T + 6.85e3iT^{2} \)
23 \( 1 - 119. iT - 1.21e4T^{2} \)
29 \( 1 + (78.5 + 78.5i)T + 2.43e4iT^{2} \)
31 \( 1 + 200. iT - 2.97e4T^{2} \)
37 \( 1 + (-76.9 - 76.9i)T + 5.06e4iT^{2} \)
41 \( 1 + 279.T + 6.89e4T^{2} \)
43 \( 1 + (15.7 - 15.7i)T - 7.95e4iT^{2} \)
47 \( 1 - 470.T + 1.03e5T^{2} \)
53 \( 1 + (-112. + 112. i)T - 1.48e5iT^{2} \)
59 \( 1 + (241. + 241. i)T + 2.05e5iT^{2} \)
61 \( 1 + (6.00 - 6.00i)T - 2.26e5iT^{2} \)
67 \( 1 + (273. + 273. i)T + 3.00e5iT^{2} \)
71 \( 1 + 448. iT - 3.57e5T^{2} \)
73 \( 1 + 54.7iT - 3.89e5T^{2} \)
79 \( 1 - 29.1iT - 4.93e5T^{2} \)
83 \( 1 + (-893. + 893. i)T - 5.71e5iT^{2} \)
89 \( 1 + 281.T + 7.04e5T^{2} \)
97 \( 1 - 188.T + 9.12e5T^{2} \)
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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

−9.331781029643938217093941177345, −8.280306891940406222978229043322, −7.60292198423394446099663017176, −7.16365298892635702633245961007, −6.13368226911150773566973940738, −4.85133272852088522084246367793, −4.08105022089827652772899989832, −3.17427177931676202870145896936, −2.05762537542030553041584555329, −0.48161554560026414204515325838, 0.974348069947086184912089064533, 1.71682187183149770881729408752, 3.70412140646340635190686902083, 4.22261794932553472618770127087, 4.98857268820425793133750704335, 6.00444366765961887921566621526, 7.20687545483440139040442176838, 8.191547501151032632337511758877, 8.536068491594335090633736907032, 8.989420390905668989116006399362

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