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Properties

Label 2-288-288.83-c1-0-23
Degree $2$
Conductor $288$
Sign $0.784 - 0.620i$
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  + (−1.12 + 0.853i)2-s + (1.31 + 1.12i)3-s + (0.543 − 1.92i)4-s + (1.72 − 2.24i)5-s + (−2.44 − 0.145i)6-s + (−1.56 + 0.419i)7-s + (1.03 + 2.63i)8-s + (0.467 + 2.96i)9-s + (−0.0265 + 4.00i)10-s + (3.02 + 0.398i)11-s + (2.88 − 1.92i)12-s + (−0.585 − 4.44i)13-s + (1.40 − 1.81i)14-s + (4.79 − 1.01i)15-s + (−3.40 − 2.09i)16-s + 7.55·17-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)2-s + (0.760 + 0.649i)3-s + (0.271 − 0.962i)4-s + (0.771 − 1.00i)5-s + (−0.998 − 0.0592i)6-s + (−0.592 + 0.158i)7-s + (0.364 + 0.931i)8-s + (0.155 + 0.987i)9-s + (−0.00838 + 1.26i)10-s + (0.912 + 0.120i)11-s + (0.831 − 0.555i)12-s + (−0.162 − 1.23i)13-s + (0.376 − 0.483i)14-s + (1.23 − 0.262i)15-s + (−0.852 − 0.522i)16-s + 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.784 - 0.620i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.784 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19696 + 0.415872i\)
\(L(\frac12)\) \(\approx\) \(1.19696 + 0.415872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.853i)T \)
3 \( 1 + (-1.31 - 1.12i)T \)
good5 \( 1 + (-1.72 + 2.24i)T + (-1.29 - 4.82i)T^{2} \)
7 \( 1 + (1.56 - 0.419i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.02 - 0.398i)T + (10.6 + 2.84i)T^{2} \)
13 \( 1 + (0.585 + 4.44i)T + (-12.5 + 3.36i)T^{2} \)
17 \( 1 - 7.55T + 17T^{2} \)
19 \( 1 + (0.152 - 0.367i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.33 - 4.98i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.840 - 0.644i)T + (7.50 - 28.0i)T^{2} \)
31 \( 1 + (0.703 - 0.406i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.96 - 2.05i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.36 + 2.24i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.719 + 5.46i)T + (-41.5 - 11.1i)T^{2} \)
47 \( 1 + (2.18 + 1.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.89 + 3.27i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.48 + 3.44i)T + (15.2 + 56.9i)T^{2} \)
61 \( 1 + (3.22 + 4.20i)T + (-15.7 + 58.9i)T^{2} \)
67 \( 1 + (-2.04 - 15.5i)T + (-64.7 + 17.3i)T^{2} \)
71 \( 1 + (8.83 - 8.83i)T - 71iT^{2} \)
73 \( 1 + (1.31 + 1.31i)T + 73iT^{2} \)
79 \( 1 + (-5.85 + 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.16 + 6.26i)T + (21.4 - 80.1i)T^{2} \)
89 \( 1 + (11.3 + 11.3i)T + 89iT^{2} \)
97 \( 1 + (-0.937 + 1.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.88333403271581067890240800815, −10.24541604322519639648157867481, −9.886158970245050405144211067052, −9.088653136799849050706543458383, −8.332956076732421466122008808452, −7.32188378955462661246399352087, −5.76089128097029932883752088875, −5.16470554252852233548365839463, −3.37934468181094404237744015330, −1.53426328610716543704312510044, 1.60228719689006088004472663673, 2.82984753866621732627225811710, 3.78593055795118548326150546900, 6.38791283734524865952879273648, 6.84265318303262096588751570525, 7.930719360336822642595727212277, 9.123510200804957502510933704828, 9.714558943888058283612978734770, 10.51774817316508058533736390662, 11.82244759745349710110377593509

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