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Properties

Label 2-288-8.5-c3-0-11
Degree $2$
Conductor $288$
Sign $-0.829 + 0.559i$
Analytic cond. $16.9925$
Root an. cond. $4.12220$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.32i·5-s − 10·7-s − 37.9i·11-s + 59.3i·13-s − 75.0·17-s − 118. i·19-s − 150.·23-s + 85·25-s − 246. i·29-s − 62·31-s − 63.2i·35-s + 59.3i·37-s − 375.·41-s + 118. i·43-s − 450.·47-s + ⋯
L(s)  = 1  + 0.565i·5-s − 0.539·7-s − 1.04i·11-s + 1.26i·13-s − 1.07·17-s − 1.43i·19-s − 1.36·23-s + 0.680·25-s − 1.57i·29-s − 0.359·31-s − 0.305i·35-s + 0.263i·37-s − 1.42·41-s + 0.420i·43-s − 1.39·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.559i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.829 + 0.559i$
Analytic conductor: \(16.9925\)
Root analytic conductor: \(4.12220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :3/2),\ -0.829 + 0.559i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3267886944\)
\(L(\frac12)\) \(\approx\) \(0.3267886944\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.32iT - 125T^{2} \)
7 \( 1 + 10T + 343T^{2} \)
11 \( 1 + 37.9iT - 1.33e3T^{2} \)
13 \( 1 - 59.3iT - 2.19e3T^{2} \)
17 \( 1 + 75.0T + 4.91e3T^{2} \)
19 \( 1 + 118. iT - 6.85e3T^{2} \)
23 \( 1 + 150.T + 1.21e4T^{2} \)
29 \( 1 + 246. iT - 2.43e4T^{2} \)
31 \( 1 + 62T + 2.97e4T^{2} \)
37 \( 1 - 59.3iT - 5.06e4T^{2} \)
41 \( 1 + 375.T + 6.89e4T^{2} \)
43 \( 1 - 118. iT - 7.95e4T^{2} \)
47 \( 1 + 450.T + 1.03e5T^{2} \)
53 \( 1 + 132. iT - 1.48e5T^{2} \)
59 \( 1 + 733. iT - 2.05e5T^{2} \)
61 \( 1 + 533. iT - 2.26e5T^{2} \)
67 \( 1 - 711. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 30T + 3.89e5T^{2} \)
79 \( 1 + 94T + 4.93e5T^{2} \)
83 \( 1 - 670. iT - 5.71e5T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 - 130T + 9.12e5T^{2} \)
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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

−11.25371417326831635815340307304, −10.01297826558520327542894029339, −9.132631268427274817199033464170, −8.182955577806507176805367752384, −6.73018309164840051319366687763, −6.35275359434611735562757618653, −4.74174444524037927031743085510, −3.49911373794932381001709001421, −2.22932242136234250296507859662, −0.11598757161119102119435961137, 1.72726583032781484984080730914, 3.35143552432572228030071450784, 4.63733058652809752898203247451, 5.71787271307614744210991680586, 6.87359066905063971362898920330, 7.973105939005372437240550521558, 8.889061978815088400324486817702, 9.984453758641049395333518744955, 10.57792832053785581881922710506, 12.03052077225610577370585825443

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