Properties

Label 2-1152-12.11-c1-0-3
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  − 3.41i·5-s + 4.82i·7-s − 2.82·11-s + 2.82·13-s + 5.41i·17-s + 5.65i·19-s + 1.17·23-s − 6.65·25-s − 0.585i·29-s − 3.17i·31-s + 16.4·35-s + 3.65·37-s + 2.58i·41-s + 9.65i·43-s + 12.4·47-s + ⋯
L(s)  = 1  − 1.52i·5-s + 1.82i·7-s − 0.852·11-s + 0.784·13-s + 1.31i·17-s + 1.29i·19-s + 0.244·23-s − 1.33·25-s − 0.108i·29-s − 0.569i·31-s + 2.78·35-s + 0.601·37-s + 0.403i·41-s + 1.47i·43-s + 1.82·47-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: $\chi_{1152} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.394543776\)
\(L(\frac12)\) \(\approx\) \(1.394543776\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.41iT - 5T^{2} \)
7 \( 1 - 4.82iT - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 5.41iT - 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 + 0.585iT - 29T^{2} \)
31 \( 1 + 3.17iT - 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 - 9.65iT - 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 5.07iT - 53T^{2} \)
59 \( 1 - 2.34T + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 6.48iT - 79T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−9.659782263365283782899747319839, −9.006824808807019217940217507063, −8.265305353930769001933405575848, −7.989776243437302024940850642402, −6.07729898028013508789540023631, −5.77471564526972110438194069593, −4.92732105823723651188326058489, −3.87139355916767467356030409684, −2.46970747226928753047365743098, −1.38847680784358031150558796350, 0.64734792517724004595993496822, 2.53141991830990794282502796096, 3.36945640649224550823612231383, 4.29028695802397280129765324100, 5.42205513459670754041838068018, 6.76024066656569152880103174150, 7.07957111567331679685021720726, 7.67198198379906614342749987224, 8.890880769226326988577320052968, 10.01331724837133053638583730926

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