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Properties

Label 2-1152-24.5-c2-0-26
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  + 5.83·5-s + 2.82·7-s + 16.4·11-s − 8.24i·13-s − 7.07i·17-s − 23.3i·19-s − 20i·23-s + 9·25-s − 29.1·29-s + 42.4·31-s + 16.4·35-s + 49.4i·37-s + 26.8i·41-s − 44i·47-s − 41·49-s + ⋯
L(s)  = 1  + 1.16·5-s + 0.404·7-s + 1.49·11-s − 0.634i·13-s − 0.415i·17-s − 1.22i·19-s − 0.869i·23-s + 0.359·25-s − 1.00·29-s + 1.36·31-s + 0.471·35-s + 1.33i·37-s + 0.655i·41-s − 0.936i·47-s − 0.836·49-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: $\chi_{1152} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.801605181\)
\(L(\frac12)\) \(\approx\) \(2.801605181\)
\(L(2)\) 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 \( 1 - 5.83T + 25T^{2} \)
7 \( 1 - 2.82T + 49T^{2} \)
11 \( 1 - 16.4T + 121T^{2} \)
13 \( 1 + 8.24iT - 169T^{2} \)
17 \( 1 + 7.07iT - 289T^{2} \)
19 \( 1 + 23.3iT - 361T^{2} \)
23 \( 1 + 20iT - 529T^{2} \)
29 \( 1 + 29.1T + 841T^{2} \)
31 \( 1 - 42.4T + 961T^{2} \)
37 \( 1 - 49.4iT - 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + 44iT - 2.20e3T^{2} \)
53 \( 1 - 29.1T + 2.80e3T^{2} \)
59 \( 1 + 65.9T + 3.48e3T^{2} \)
61 \( 1 + 82.4iT - 3.72e3T^{2} \)
67 \( 1 - 116. iT - 4.48e3T^{2} \)
71 \( 1 + 100iT - 5.04e3T^{2} \)
73 \( 1 - 40T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 + 82.4T + 6.88e3T^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 + 40T + 9.40e3T^{2} \)
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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

−9.529345730384669745127367689305, −8.866889999255163918176478552896, −7.979661527951777801806091955463, −6.74423129211863667815179233469, −6.31986976443721376621950445936, −5.23585963851188870884412810533, −4.45889341373638384729021588452, −3.13002595626263879655497781263, −2.03680618682403612314357916338, −0.924892892703000040856471765640, 1.38513249592748879950850489441, 2.00946055426531084801465804490, 3.57754217924805269470057020107, 4.42891203214250397216455062954, 5.70499440247391665211769802908, 6.14810167232496690830413616052, 7.10445465999912801318808590268, 8.104612227194808734244939274988, 9.144642242576143338234061297348, 9.518537165027840802010710772839

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