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Properties

Label 2-1152-16.5-c1-0-9
Degree $2$
Conductor $1152$
Sign $0.983 + 0.179i$
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  + (2.37 + 2.37i)5-s − 3.64i·7-s + (0.841 + 0.841i)11-s + (2.64 − 2.64i)13-s + 3.06·17-s + (−1.64 + 1.64i)19-s − 7.82i·23-s + 6.29i·25-s + (0.692 − 0.692i)29-s − 0.354·31-s + (8.66 − 8.66i)35-s + (−4.64 − 4.64i)37-s + 6.43i·41-s + (5.64 + 5.64i)43-s + 11.1·47-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)5-s − 1.37i·7-s + (0.253 + 0.253i)11-s + (0.733 − 0.733i)13-s + 0.744·17-s + (−0.377 + 0.377i)19-s − 1.63i·23-s + 1.25i·25-s + (0.128 − 0.128i)29-s − 0.0636·31-s + (1.46 − 1.46i)35-s + (−0.763 − 0.763i)37-s + 1.00i·41-s + (0.860 + 0.860i)43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.091185388\)
\(L(\frac12)\) \(\approx\) \(2.091185388\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 - 2.37i)T + 5iT^{2} \)
7 \( 1 + 3.64iT - 7T^{2} \)
11 \( 1 + (-0.841 - 0.841i)T + 11iT^{2} \)
13 \( 1 + (-2.64 + 2.64i)T - 13iT^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 + (1.64 - 1.64i)T - 19iT^{2} \)
23 \( 1 + 7.82iT - 23T^{2} \)
29 \( 1 + (-0.692 + 0.692i)T - 29iT^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 + (4.64 + 4.64i)T + 37iT^{2} \)
41 \( 1 - 6.43iT - 41T^{2} \)
43 \( 1 + (-5.64 - 5.64i)T + 43iT^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + (-5.44 - 5.44i)T + 53iT^{2} \)
59 \( 1 + (7.82 + 7.82i)T + 59iT^{2} \)
61 \( 1 + (4.64 - 4.64i)T - 61iT^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + 7.29iT - 73T^{2} \)
79 \( 1 - 4.35T + 79T^{2} \)
83 \( 1 + (0.841 - 0.841i)T - 83iT^{2} \)
89 \( 1 - 9.50iT - 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.06141479572024928423599655568, −9.103278107749451767595943669311, −7.966108682253854894586244635894, −7.21651058029666449007389868332, −6.39559722850177755347878017309, −5.81569473991081324604570431280, −4.46222778699970940274198437459, −3.51012381257332033258149728583, −2.47946067490869324333941794392, −1.09372773694661418145322600136, 1.35578242007852919516161543336, 2.24744782978666888237340863287, 3.61494759586195468539558278528, 4.91175133040684829610271522287, 5.68526762634222844304551854028, 6.07567267439158258420739182431, 7.35394254366910525335200309735, 8.681779004211610917817690481041, 8.906276089755782033355933339517, 9.530079434226423377326306614952

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