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Properties

Label 2-1152-24.11-c1-0-5
Degree $2$
Conductor $1152$
Sign $0.577 - 0.816i$
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  + 1.41·5-s + 6i·13-s + 4.24i·17-s − 2.99·25-s + 9.89·29-s + 12i·37-s − 12.7i·41-s + 7·49-s + 7.07·53-s + 12i·61-s + 8.48i·65-s − 16·73-s + 6i·85-s − 4.24i·89-s + 8·97-s + ⋯
L(s)  = 1  + 0.632·5-s + 1.66i·13-s + 1.02i·17-s − 0.599·25-s + 1.83·29-s + 1.97i·37-s − 1.98i·41-s + 49-s + 0.971·53-s + 1.53i·61-s + 1.05i·65-s − 1.87·73-s + 0.650i·85-s − 0.449i·89-s + 0.812·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724137031\)
\(L(\frac12)\) \(\approx\) \(1.724137031\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12iT - 37T^{2} \)
41 \( 1 + 12.7iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.09120346222069976960340433905, −8.974116559826398439835351954084, −8.548416245602243956283442436278, −7.32245350535378025335429001549, −6.51326974067145458704143866011, −5.84475588624455757937656780684, −4.68759441609376948252075447161, −3.87559949350476575447948535690, −2.47993195824009665682586058296, −1.47576052914384696253817129458, 0.797260479255646131653182496628, 2.38787260501486819090338192923, 3.25300848910838711270412377231, 4.60311563185912939554430170615, 5.49324493975543760523002417614, 6.17180035881777150864930677334, 7.25956749918183102744458763751, 8.015217658676584495778404903220, 8.889812951817189250062383922438, 9.819362165671936821287084924066

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