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Properties

Label 2-1152-8.5-c1-0-2
Degree $2$
Conductor $1152$
Sign $-0.707 - 0.707i$
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  + 2i·5-s + 4i·13-s − 8·17-s + 25-s + 10i·29-s − 12i·37-s − 8·41-s − 7·49-s + 14i·53-s + 12i·61-s − 8·65-s − 6·73-s − 16i·85-s − 16·89-s + 18·97-s + ⋯
L(s)  = 1  + 0.894i·5-s + 1.10i·13-s − 1.94·17-s + 0.200·25-s + 1.85i·29-s − 1.97i·37-s − 1.24·41-s − 49-s + 1.92i·53-s + 1.53i·61-s − 0.992·65-s − 0.702·73-s − 1.73i·85-s − 1.69·89-s + 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9672612292\)
\(L(\frac12)\) \(\approx\) \(0.9672612292\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−10.26004222360917259799611803331, −9.038896818081904061207915429151, −8.793481022831512363938211169209, −7.31300450818656449038789296103, −6.89321737430630301621976437192, −6.10686217759843019636466558994, −4.86225409412988049317034377156, −3.98969828388277430097709521455, −2.85131357734068010185036645810, −1.82404577004519249888526111108, 0.39953889036226379155252783473, 1.92655968857620225165092396287, 3.18575233577633167701919505878, 4.46573501861109004967213060939, 5.02848065005896126532311505996, 6.13629272810746063422679438787, 6.91615205120311970156970765799, 8.263873305908931416619055964499, 8.392240848196823240102642816951, 9.526389366691702526219244669729

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