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Properties

Label 2-1152-16.5-c1-0-14
Degree $2$
Conductor $1152$
Sign $-0.270 + 0.962i$
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  + (−0.334 − 0.334i)5-s − 4.55i·7-s + (2.47 + 2.47i)11-s + (0.0594 − 0.0594i)13-s − 3.61·17-s + (2.55 − 2.55i)19-s − 2.82i·23-s − 4.77i·25-s + (−5.16 + 5.16i)29-s + 0.557·31-s + (−1.52 + 1.52i)35-s + (−4.38 − 4.38i)37-s − 9.27i·41-s + (−1.61 − 1.61i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (−0.149 − 0.149i)5-s − 1.72i·7-s + (0.745 + 0.745i)11-s + (0.0164 − 0.0164i)13-s − 0.877·17-s + (0.586 − 0.586i)19-s − 0.589i·23-s − 0.955i·25-s + (−0.958 + 0.958i)29-s + 0.100·31-s + (−0.258 + 0.258i)35-s + (−0.721 − 0.721i)37-s − 1.44i·41-s + (−0.245 − 0.245i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.270 + 0.962i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.291478859\)
\(L(\frac12)\) \(\approx\) \(1.291478859\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.334 + 0.334i)T + 5iT^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 + (-2.47 - 2.47i)T + 11iT^{2} \)
13 \( 1 + (-0.0594 + 0.0594i)T - 13iT^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + (-2.55 + 2.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (5.16 - 5.16i)T - 29iT^{2} \)
31 \( 1 - 0.557T + 31T^{2} \)
37 \( 1 + (4.38 + 4.38i)T + 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (1.61 + 1.61i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (0.493 + 0.493i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (2.72 - 2.72i)T - 61iT^{2} \)
67 \( 1 + (-3.77 + 3.77i)T - 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 - 0.541iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−9.520858490780498706321182083417, −8.860356379965984476641709628317, −7.71747136170142585733181392906, −7.06286471870003836577271209837, −6.51021005101125704788205357453, −5.03616756785209964365924449069, −4.26087299736389811290514798841, −3.55644915261042393834848244146, −1.94517964183321982380805126794, −0.56703357513918046513125123159, 1.64854184242485776290998916874, 2.83147404710934290085600512635, 3.75732279203597202039562600613, 5.07607058069834984791505567065, 5.86227678497890758329708703794, 6.50514133034329217093258936248, 7.69383452353418717657722151502, 8.531272168960200608656199974212, 9.189002486734100139141242819448, 9.772357509111399676319486338056

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