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Properties

Label 2-1152-3.2-c2-0-13
Degree $2$
Conductor $1152$
Sign $0.816 + 0.577i$
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  + 4.87i·5-s − 11.7·7-s − 9.75i·11-s − 22.6·13-s + 22.1i·17-s + 17.7·19-s − 14.1i·23-s + 1.20·25-s − 20.0i·29-s + 39.7·31-s − 57.5i·35-s − 2.40·37-s − 64.3i·41-s − 3.19·43-s − 41.8i·47-s + ⋯
L(s)  = 1  + 0.975i·5-s − 1.68·7-s − 0.886i·11-s − 1.74·13-s + 1.30i·17-s + 0.936·19-s − 0.614i·23-s + 0.0480·25-s − 0.689i·29-s + 1.28·31-s − 1.64i·35-s − 0.0649·37-s − 1.56i·41-s − 0.0742·43-s − 0.890i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9761303077\)
\(L(\frac12)\) \(\approx\) \(0.9761303077\)
\(L(2)\) 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 \( 1 - 4.87iT - 25T^{2} \)
7 \( 1 + 11.7T + 49T^{2} \)
11 \( 1 + 9.75iT - 121T^{2} \)
13 \( 1 + 22.6T + 169T^{2} \)
17 \( 1 - 22.1iT - 289T^{2} \)
19 \( 1 - 17.7T + 361T^{2} \)
23 \( 1 + 14.1iT - 529T^{2} \)
29 \( 1 + 20.0iT - 841T^{2} \)
31 \( 1 - 39.7T + 961T^{2} \)
37 \( 1 + 2.40T + 1.36e3T^{2} \)
41 \( 1 + 64.3iT - 1.68e3T^{2} \)
43 \( 1 + 3.19T + 1.84e3T^{2} \)
47 \( 1 + 41.8iT - 2.20e3T^{2} \)
53 \( 1 - 55.5iT - 2.80e3T^{2} \)
59 \( 1 - 111. iT - 3.48e3T^{2} \)
61 \( 1 + 10.8T + 3.72e3T^{2} \)
67 \( 1 - 18.2T + 4.48e3T^{2} \)
71 \( 1 + 34.7iT - 5.04e3T^{2} \)
73 \( 1 - 87.5T + 5.32e3T^{2} \)
79 \( 1 - 151.T + 6.24e3T^{2} \)
83 \( 1 + 61.8iT - 6.88e3T^{2} \)
89 \( 1 - 72.9iT - 7.92e3T^{2} \)
97 \( 1 + 87.3T + 9.40e3T^{2} \)
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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

−9.732022934131313518966912655041, −8.822214306974203960944810899123, −7.72322216296373207409974032980, −6.88961183828961391015109324318, −6.32670867689445088743032657016, −5.47678830966665186965807040883, −4.05026804549170209153675096445, −3.10426346327897135685863972591, −2.50647276578129634348582102298, −0.40491247981270283916694061408, 0.804914850890778932411711137567, 2.48327639166800387397039894368, 3.35038757233585790783861980824, 4.79847217933596630385796533873, 5.11811899401695851129424763336, 6.50223555695454273667829709796, 7.14135658380713817865548926889, 7.953269654431238855321134631650, 9.287186610289203905331822027353, 9.627726135647207689778424232365

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