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Properties

Label 2-1152-48.5-c2-0-15
Degree $2$
Conductor $1152$
Sign $0.585 - 0.810i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
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  + (−5.14 + 5.14i)5-s + 7.48i·7-s + (11.6 − 11.6i)11-s + (14.0 − 14.0i)13-s + 7.92i·17-s + (10.7 − 10.7i)19-s − 2.09·23-s − 27.9i·25-s + (−23.6 − 23.6i)29-s + 17.8·31-s + (−38.5 − 38.5i)35-s + (30.8 + 30.8i)37-s + 36.8·41-s + (28.4 + 28.4i)43-s + 65.3i·47-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)5-s + 1.06i·7-s + (1.06 − 1.06i)11-s + (1.08 − 1.08i)13-s + 0.466i·17-s + (0.567 − 0.567i)19-s − 0.0909·23-s − 1.11i·25-s + (−0.814 − 0.814i)29-s + 0.576·31-s + (−1.10 − 1.10i)35-s + (0.834 + 0.834i)37-s + 0.898·41-s + (0.660 + 0.660i)43-s + 1.39i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.739346898\)
\(L(\frac12)\) \(\approx\) \(1.739346898\)
\(L(2)\) 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 \( 1 + (5.14 - 5.14i)T - 25iT^{2} \)
7 \( 1 - 7.48iT - 49T^{2} \)
11 \( 1 + (-11.6 + 11.6i)T - 121iT^{2} \)
13 \( 1 + (-14.0 + 14.0i)T - 169iT^{2} \)
17 \( 1 - 7.92iT - 289T^{2} \)
19 \( 1 + (-10.7 + 10.7i)T - 361iT^{2} \)
23 \( 1 + 2.09T + 529T^{2} \)
29 \( 1 + (23.6 + 23.6i)T + 841iT^{2} \)
31 \( 1 - 17.8T + 961T^{2} \)
37 \( 1 + (-30.8 - 30.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 36.8T + 1.68e3T^{2} \)
43 \( 1 + (-28.4 - 28.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 65.3iT - 2.20e3T^{2} \)
53 \( 1 + (-9.05 + 9.05i)T - 2.80e3iT^{2} \)
59 \( 1 + (74.0 - 74.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-53.4 + 53.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (20.6 - 20.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 39.6T + 5.04e3T^{2} \)
73 \( 1 - 91.3iT - 5.32e3T^{2} \)
79 \( 1 + 92.1T + 6.24e3T^{2} \)
83 \( 1 + (-9.12 - 9.12i)T + 6.88e3iT^{2} \)
89 \( 1 - 63.2T + 7.92e3T^{2} \)
97 \( 1 - 152.T + 9.40e3T^{2} \)
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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

−9.667984808438723063057559216170, −8.754902739916419049465736657031, −8.139653341514383162673737209013, −7.34646519159761751678798282069, −6.04672298353936629282636457103, −5.98142530107961759638362865602, −4.30380994545191990192270728009, −3.37299898789112819267375897172, −2.76634742229827579544651427747, −0.945953903415433834649082776872, 0.74423017851853489815427015389, 1.67477186808995163796802363948, 3.76109636401653365449403317902, 4.07357259264707833411454576073, 4.89468340024096765189185775809, 6.23237617505910917009952986027, 7.23970257575469892515433147595, 7.65597926927277798090133342482, 8.851115419333871630731444751030, 9.232717508156633551147979990337

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