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Properties

Label 2-288-8.5-c1-0-3
Degree $2$
Conductor $288$
Sign $i$
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  − 2.82i·5-s − 2·7-s − 5.65i·11-s − 3.00·25-s − 2.82i·29-s + 10·31-s + 5.65i·35-s − 3·49-s + 14.1i·53-s − 16.0·55-s + 11.3i·59-s + 14·73-s + 11.3i·77-s + 10·79-s − 5.65i·83-s + ⋯
L(s)  = 1  − 1.26i·5-s − 0.755·7-s − 1.70i·11-s − 0.600·25-s − 0.525i·29-s + 1.79·31-s + 0.956i·35-s − 0.428·49-s + 1.94i·53-s − 2.15·55-s + 1.47i·59-s + 1.63·73-s + 1.28i·77-s + 1.12·79-s − 0.620i·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766762 - 0.766762i\)
\(L(\frac12)\) \(\approx\) \(0.766762 - 0.766762i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.1iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.72416379557190329074753040666, −10.63674661622151076626270211416, −9.520030237277237139040140729588, −8.723941396503243354537087746867, −7.989377706501535148655064055663, −6.42453697861332577235081388122, −5.58852923218766745164308619075, −4.34660200005802142189931412234, −3.01699593577955395555041788775, −0.838941618298207999531843342580, 2.31131467725737623642794538331, 3.49605654314827524359638190567, 4.88202694461564893056033644935, 6.50007918929654233675800096403, 6.91634917651026700944572235287, 8.043749302314307753779642083378, 9.586479804041296252602956632838, 10.05913928040055511369463331455, 11.00723200847019826061320292563, 12.08278025272979029846000667134

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