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Properties

Label 2-1152-16.5-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.179 + 0.983i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (−0.595 − 0.595i)5-s + 1.64i·7-s + (−3.36 − 3.36i)11-s + (−2.64 + 2.64i)13-s + 5.53·17-s + (3.64 − 3.64i)19-s − 4.33i·23-s − 4.29i·25-s + (6.12 − 6.12i)29-s − 5.64·31-s + (0.979 − 0.979i)35-s + (0.645 + 0.645i)37-s − 7.91i·41-s + (0.354 + 0.354i)43-s − 9.10·47-s + ⋯
L(s)  = 1  + (−0.266 − 0.266i)5-s + 0.622i·7-s + (−1.01 − 1.01i)11-s + (−0.733 + 0.733i)13-s + 1.34·17-s + (0.836 − 0.836i)19-s − 0.904i·23-s − 0.858i·25-s + (1.13 − 1.13i)29-s − 1.01·31-s + (0.165 − 0.165i)35-s + (0.106 + 0.106i)37-s − 1.23i·41-s + (0.0540 + 0.0540i)43-s − 1.32·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.179 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.165337695\)
\(L(\frac12)\) \(\approx\) \(1.165337695\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.595 + 0.595i)T + 5iT^{2} \)
7 \( 1 - 1.64iT - 7T^{2} \)
11 \( 1 + (3.36 + 3.36i)T + 11iT^{2} \)
13 \( 1 + (2.64 - 2.64i)T - 13iT^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 + (-3.64 + 3.64i)T - 19iT^{2} \)
23 \( 1 + 4.33iT - 23T^{2} \)
29 \( 1 + (-6.12 + 6.12i)T - 29iT^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 + (-0.645 - 0.645i)T + 37iT^{2} \)
41 \( 1 + 7.91iT - 41T^{2} \)
43 \( 1 + (-0.354 - 0.354i)T + 43iT^{2} \)
47 \( 1 + 9.10T + 47T^{2} \)
53 \( 1 + (-4.93 - 4.93i)T + 53iT^{2} \)
59 \( 1 + (4.33 + 4.33i)T + 59iT^{2} \)
61 \( 1 + (-0.645 + 0.645i)T - 61iT^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 - 3.29iT - 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 + (-3.36 + 3.36i)T - 83iT^{2} \)
89 \( 1 + 2.38iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−9.600725728022093626937379290191, −8.718229336788901390959578797829, −8.057967443242239726272991684887, −7.25585558197197895257910530685, −6.12863158883867184292252997127, −5.33603015280673683353357518938, −4.53892602156968182495961475270, −3.20692335155182357171492526169, −2.35504345765241218341822881802, −0.53683608020048707484273994794, 1.36299635298279026484515446661, 2.90177950710938501959687439580, 3.67211788129810419324161558776, 5.03135932801939314522165327587, 5.48158726033763709380385567253, 6.92322860749254641753222862415, 7.65738168992480184827250359228, 7.910765754776407392054099871100, 9.402937322772131710255023945856, 10.10252024609195352182465831053

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