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Properties

Label 2-288-32.29-c1-0-5
Degree $2$
Conductor $288$
Sign $-0.437 - 0.899i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (0.607 + 1.27i)2-s + (−1.26 + 1.55i)4-s + (1.35 + 3.27i)5-s + (2.48 − 2.48i)7-s + (−2.74 − 0.668i)8-s + (−3.36 + 3.72i)10-s + (−0.420 + 0.174i)11-s + (−1.98 + 4.80i)13-s + (4.68 + 1.66i)14-s + (−0.816 − 3.91i)16-s − 4.75i·17-s + (−0.402 + 0.971i)19-s + (−6.80 − 2.02i)20-s + (−0.478 − 0.431i)22-s + (−0.739 − 0.739i)23-s + ⋯
L(s)  = 1  + (0.429 + 0.903i)2-s + (−0.630 + 0.775i)4-s + (0.607 + 1.46i)5-s + (0.939 − 0.939i)7-s + (−0.971 − 0.236i)8-s + (−1.06 + 1.17i)10-s + (−0.126 + 0.0525i)11-s + (−0.551 + 1.33i)13-s + (1.25 + 0.444i)14-s + (−0.204 − 0.978i)16-s − 1.15i·17-s + (−0.0923 + 0.222i)19-s + (−1.52 − 0.453i)20-s + (−0.101 − 0.0919i)22-s + (−0.154 − 0.154i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.437 - 0.899i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881961 + 1.41055i\)
\(L(\frac12)\) \(\approx\) \(0.881961 + 1.41055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.607 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (-1.35 - 3.27i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.48 + 2.48i)T - 7iT^{2} \)
11 \( 1 + (0.420 - 0.174i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.98 - 4.80i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 4.75iT - 17T^{2} \)
19 \( 1 + (0.402 - 0.971i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.739 + 0.739i)T + 23iT^{2} \)
29 \( 1 + (-0.153 - 0.0634i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 8.57T + 31T^{2} \)
37 \( 1 + (2.67 + 6.46i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.39 - 1.39i)T + 41iT^{2} \)
43 \( 1 + (-2.84 + 1.18i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.715iT - 47T^{2} \)
53 \( 1 + (-10.4 + 4.32i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.09 + 5.05i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.81 + 1.16i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.39 + 2.23i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-8.26 + 8.26i)T - 71iT^{2} \)
73 \( 1 + (-4.37 - 4.37i)T + 73iT^{2} \)
79 \( 1 + 9.46iT - 79T^{2} \)
83 \( 1 + (2.85 - 6.88i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.60 - 8.60i)T - 89iT^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−12.06267798754880839953847306623, −11.21943268932343874711796254894, −10.21462389680189672272416377349, −9.260160804289026925334636510360, −7.83964540496268168366452116022, −7.08255819106397994924650962518, −6.46221157320570942301104542444, −5.06785638466797040939387647194, −4.01123927541272156086845833434, −2.49364841428214315711327521680, 1.29196913994044020028879638901, 2.57730611164913587448642380549, 4.42737954545607721184092108317, 5.28374349730671459645574898458, 5.87831576290856005712923289946, 8.215792703355600506724120299742, 8.656655919223750654386650592120, 9.753650697745379794838839131459, 10.57436519521214105490315019253, 11.81588338017555107538312184636

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