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Properties

Label 2-1152-72.59-c1-0-22
Degree $2$
Conductor $1152$
Sign $0.635 - 0.771i$
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  + (1.57 + 0.724i)3-s + (1.94 + 2.28i)9-s + (−0.476 + 0.275i)11-s + 2.36i·17-s + 5.97·19-s + (2.5 + 4.33i)25-s + (1.41 + 5.00i)27-s + (−0.949 + 0.0874i)33-s + (9.39 + 5.42i)41-s + (−2.20 − 3.82i)43-s + (−3.5 + 6.06i)49-s + (−1.71 + 3.72i)51-s + (9.39 + 4.33i)57-s + (−13.2 − 7.62i)59-s + (8.18 − 14.1i)67-s + ⋯
L(s)  = 1  + (0.908 + 0.418i)3-s + (0.649 + 0.760i)9-s + (−0.143 + 0.0829i)11-s + 0.574i·17-s + 1.37·19-s + (0.5 + 0.866i)25-s + (0.272 + 0.962i)27-s + (−0.165 + 0.0152i)33-s + (1.46 + 0.847i)41-s + (−0.336 − 0.583i)43-s + (−0.5 + 0.866i)49-s + (−0.240 + 0.521i)51-s + (1.24 + 0.573i)57-s + (−1.71 − 0.992i)59-s + (0.999 − 1.73i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.635 - 0.771i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.300123603\)
\(L(\frac12)\) \(\approx\) \(2.300123603\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 0.724i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.476 - 0.275i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.36iT - 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-9.39 - 5.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.20 + 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (13.2 + 7.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 + 14.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.5 - 9i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (4.84 + 8.39i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.571494639610330251229880559892, −9.353818652291032384045690074543, −8.196114210230298954171068930626, −7.68243233113176325532141104354, −6.73109571147427992149091890624, −5.51413959007195103380117414036, −4.65703816589287967013136897060, −3.60905344942569948552336970690, −2.80931741383333954480511397684, −1.49957804264462971910686255714, 1.01823845388743465153627281204, 2.42448001308019327104253529481, 3.25269536384947346612010067438, 4.33286671966794267699294914555, 5.43228268516983278580398916886, 6.53178642651605443969693898930, 7.34867797583922500520659802644, 7.988494291147488627777172710412, 8.877874230336528661065872226874, 9.541771821618554955991375895789

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