Properties

Label 1-288-288.229-r0-0-0
Degree $1$
Conductor $288$
Sign $-0.402 - 0.915i$
Analytic cond. $1.33746$
Root an. cond. $1.33746$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)5-s + (−0.866 − 0.5i)7-s + (0.965 − 0.258i)11-s + (−0.965 − 0.258i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s + (−0.258 − 0.965i)29-s + (−0.5 − 0.866i)31-s + (−0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.866 + 0.5i)41-s + (0.965 − 0.258i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)5-s + (−0.866 − 0.5i)7-s + (0.965 − 0.258i)11-s + (−0.965 − 0.258i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s + (−0.258 − 0.965i)29-s + (−0.5 − 0.866i)31-s + (−0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.866 + 0.5i)41-s + (0.965 − 0.258i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(1.33746\)
Root analytic conductor: \(1.33746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 288,\ (0:\ ),\ -0.402 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5027265024 - 0.7704458572i\)
\(L(\frac12)\) \(\approx\) \(0.5027265024 - 0.7704458572i\)
\(L(1)\) \(\approx\) \(0.8509087761 - 0.3496759081i\)
\(L(1)\) \(\approx\) \(0.8509087761 - 0.3496759081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.258 + 0.965i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.7533469122489009673492189388, −25.04130783231654160179741904745, −24.110341516632525807718908025, −22.724607653847927692415973051599, −22.25147172550535373861394541119, −21.67679175931954344412569488964, −20.124809640291899284634699764131, −19.4708054639095256487329968821, −18.507127450914923483079784866461, −17.72942657734425160898277694785, −16.62080784244962125993809371234, −15.65584506349979586141603243531, −14.6051420530246421361351840785, −14.01605656466893099328479147866, −12.64457411198239928780362276273, −11.88925462016021759947235087422, −10.70688238392173396103151561377, −9.713983568463925915062768894047, −9.003853961398475857607963866390, −7.3762400016010593587478628549, −6.62217392729831011983232881255, −5.703425985462792057279671056414, −4.14732712492481598404291610474, −3.002596575295853640235739575429, −1.952668055937374142361235921137, 0.59009995119785543981257459840, 2.16264522818602691211911062690, 3.68311646533976148844562337407, 4.6780946954749732549715260404, 5.90385855573245790206036844575, 6.919519484719219455860412747571, 8.10670745000617449292235124071, 9.4253038710908062438500788662, 9.72147740272166204870334982635, 11.297964798427395967656688913110, 12.2338022032322125740136319755, 13.23515137407062400566445173364, 13.844831972540692432980409101926, 15.22389100423968222199217383697, 16.17649773812269385917913915026, 17.021906701469527735458056650769, 17.621614254719847856371501570767, 19.10779464725696722702813188871, 19.99356615296561360621285435430, 20.34634213462194341873267023915, 21.996945685796660346093307062441, 22.18140381068812753313945073283, 23.59746256300457701643771985388, 24.41221897177472017651919780400, 25.07002027366716289268280084624

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