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Properties

Label 2-1152-9.4-c1-0-25
Degree $2$
Conductor $1152$
Sign $0.939 + 0.342i$
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  + (−1.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (1 + 1.73i)7-s + (1.5 − 2.59i)9-s + (−2.5 − 4.33i)11-s + (2 − 3.46i)13-s − 3.46i·15-s + 17-s − 5·19-s + (−3 − 1.73i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 5.19i·27-s + (−3 − 5.19i)29-s + (7.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (0.377 + 0.654i)7-s + (0.5 − 0.866i)9-s + (−0.753 − 1.30i)11-s + (0.554 − 0.960i)13-s − 0.894i·15-s + 0.242·17-s − 1.14·19-s + (−0.654 − 0.377i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.999i·27-s + (−0.557 − 0.964i)29-s + (1.30 + 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9327810993\)
\(L(\frac12)\) \(\approx\) \(0.9327810993\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05269705339901852894008974909, −8.825993875729138970748968380421, −8.197951320930899893405917015520, −7.23224505009236497278634632527, −5.97066375407891166138338075674, −5.82184596543188458367752445307, −4.61050929935851078248876764669, −3.55006983724502924308295996107, −2.62920653369505239962300720912, −0.56048268052221173417437057464, 1.06692554332334423205375008396, 2.14979699937161635144707756177, 4.14233055175990686205372118648, 4.60471357704948884024862892975, 5.50866357116987469866208978767, 6.62414820290657301474689088788, 7.36789985313703095915963689758, 7.983430496040446578006628728183, 8.976563748980780377745863618160, 9.968646117772716200464282202599

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