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Properties

Label 2-1152-16.5-c1-0-3
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  + (1.74 + 1.74i)5-s + 2.55i·7-s + (−0.473 − 0.473i)11-s + (−2.88 + 2.88i)13-s + 6.44·17-s + (−4.55 + 4.55i)19-s − 2.82i·23-s + 1.11i·25-s + (−3.07 + 3.07i)29-s − 6.55·31-s + (−4.47 + 4.47i)35-s + (2.72 + 2.72i)37-s + 0.788i·41-s + (−0.389 − 0.389i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (0.782 + 0.782i)5-s + 0.966i·7-s + (−0.142 − 0.142i)11-s + (−0.800 + 0.800i)13-s + 1.56·17-s + (−1.04 + 1.04i)19-s − 0.589i·23-s + 0.223i·25-s + (−0.571 + 0.571i)29-s − 1.17·31-s + (−0.756 + 0.756i)35-s + (0.448 + 0.448i)37-s + 0.123i·41-s + (−0.0594 − 0.0594i)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

\(L(1)\) \(\approx\) \(1.563546249\)
\(L(\frac12)\) \(\approx\) \(1.563546249\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 - 1.74i)T + 5iT^{2} \)
7 \( 1 - 2.55iT - 7T^{2} \)
11 \( 1 + (0.473 + 0.473i)T + 11iT^{2} \)
13 \( 1 + (2.88 - 2.88i)T - 13iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (4.55 - 4.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (3.07 - 3.07i)T - 29iT^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + (-2.72 - 2.72i)T + 37iT^{2} \)
41 \( 1 - 0.788iT - 41T^{2} \)
43 \( 1 + (0.389 + 0.389i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (2.57 + 2.57i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (-4.38 + 4.38i)T - 61iT^{2} \)
67 \( 1 + (2.11 - 2.11i)T - 67iT^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 6.31T + 79T^{2} \)
83 \( 1 + (0.641 - 0.641i)T - 83iT^{2} \)
89 \( 1 - 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−10.00936953458774617023958631075, −9.357963048721471087128020585281, −8.455074030034117836241508472099, −7.53798642857861977466139738655, −6.56599645594881563417824716147, −5.86862281676980923107279039349, −5.13285148510401668670931842649, −3.76632042112774429024304748147, −2.63473547720514906849262475417, −1.84808602599519056854643366927, 0.66122273865409941711737548820, 1.96202575272733876866440601654, 3.29936214220657644191682424474, 4.45441943453657652378506441349, 5.30530177432609483766559443690, 5.99567375963208068137514995350, 7.33775073347617649987246167478, 7.69881399411613367327467647506, 8.913364368184750863627711919794, 9.583557671375827403686596846748

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