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Properties

Label 2-1152-48.11-c1-0-9
Degree $2$
Conductor $1152$
Sign $-0.0355 + 0.999i$
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  + (−2.63 + 2.63i)5-s + 0.207·7-s + (−3.66 − 3.66i)11-s + (−0.255 + 0.255i)13-s + 0.654i·17-s + (4.46 + 4.46i)19-s − 3.48i·23-s − 8.86i·25-s + (−4.33 − 4.33i)29-s − 6.16i·31-s + (−0.545 + 0.545i)35-s + (−4.39 − 4.39i)37-s + 0.0684·41-s + (5.65 − 5.65i)43-s + 9.14·47-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)5-s + 0.0783·7-s + (−1.10 − 1.10i)11-s + (−0.0708 + 0.0708i)13-s + 0.158i·17-s + (1.02 + 1.02i)19-s − 0.727i·23-s − 1.77i·25-s + (−0.805 − 0.805i)29-s − 1.10i·31-s + (−0.0921 + 0.0921i)35-s + (−0.722 − 0.722i)37-s + 0.0106·41-s + (0.862 − 0.862i)43-s + 1.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0355 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0355 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.0355 + 0.999i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.0355 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5789007990\)
\(L(\frac12)\) \(\approx\) \(0.5789007990\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 2.63i)T - 5iT^{2} \)
7 \( 1 - 0.207T + 7T^{2} \)
11 \( 1 + (3.66 + 3.66i)T + 11iT^{2} \)
13 \( 1 + (0.255 - 0.255i)T - 13iT^{2} \)
17 \( 1 - 0.654iT - 17T^{2} \)
19 \( 1 + (-4.46 - 4.46i)T + 19iT^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (4.33 + 4.33i)T + 29iT^{2} \)
31 \( 1 + 6.16iT - 31T^{2} \)
37 \( 1 + (4.39 + 4.39i)T + 37iT^{2} \)
41 \( 1 - 0.0684T + 41T^{2} \)
43 \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 + (1.51 - 1.51i)T - 53iT^{2} \)
59 \( 1 + (-2.53 - 2.53i)T + 59iT^{2} \)
61 \( 1 + (-5.46 + 5.46i)T - 61iT^{2} \)
67 \( 1 + (4.77 + 4.77i)T + 67iT^{2} \)
71 \( 1 + 5.94iT - 71T^{2} \)
73 \( 1 + 6.93iT - 73T^{2} \)
79 \( 1 - 4.72iT - 79T^{2} \)
83 \( 1 + (-4.32 + 4.32i)T - 83iT^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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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

−9.732603859638385533992005518542, −8.533964862182595295009690328861, −7.75884904566949719843282192057, −7.40214539022592817675053666172, −6.21430507871506391891039466005, −5.45787637874331465221780338635, −4.06663461591524026302782553167, −3.37376106685373293818027953646, −2.43042482182223109450486505583, −0.27205425076462948382031569495, 1.26752712745334771323611862479, 2.85779997651435541194426346110, 4.01544683468929385597313383657, 4.99640793130934978943603064040, 5.29787092720348061452797074595, 7.08522051859744727336837826530, 7.51210460417018809621236605055, 8.335707987002428210927332248926, 9.120530982914466528764033294631, 9.871049719889113682074942700485

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