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Properties

Label 1-864-864.13-r0-0-0
Degree $1$
Conductor $864$
Sign $0.926 + 0.375i$
Analytic cond. $4.01239$
Root an. cond. $4.01239$
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  + (0.819 + 0.573i)5-s + (0.642 − 0.766i)7-s + (0.573 + 0.819i)11-s + (−0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.906 + 0.422i)29-s + (0.766 − 0.642i)31-s + (0.965 − 0.258i)35-s + (−0.965 − 0.258i)37-s + (0.342 − 0.939i)41-s + (0.573 + 0.819i)43-s + (−0.766 − 0.642i)47-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)5-s + (0.642 − 0.766i)7-s + (0.573 + 0.819i)11-s + (−0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.906 + 0.422i)29-s + (0.766 − 0.642i)31-s + (0.965 − 0.258i)35-s + (−0.965 − 0.258i)37-s + (0.342 − 0.939i)41-s + (0.573 + 0.819i)43-s + (−0.766 − 0.642i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(4.01239\)
Root analytic conductor: \(4.01239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 864,\ (0:\ ),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.886561189 + 0.3681343319i\)
\(L(\frac12)\) \(\approx\) \(1.886561189 + 0.3681343319i\)
\(L(1)\) \(\approx\) \(1.351419727 + 0.1183162725i\)
\(L(1)\) \(\approx\) \(1.351419727 + 0.1183162725i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.819 + 0.573i)T \)
7 \( 1 + (0.642 - 0.766i)T \)
11 \( 1 + (0.573 + 0.819i)T \)
13 \( 1 + (-0.422 - 0.906i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.906 + 0.422i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (0.573 + 0.819i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.819 + 0.573i)T \)
61 \( 1 + (-0.0871 - 0.996i)T \)
67 \( 1 + (0.422 + 0.906i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.906 - 0.422i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−21.79559580675035047633151789384, −21.171229818950041323629950091237, −20.78923850084226858526804757913, −19.429478750965192213444718167810, −18.92469941122490722036713338128, −17.9253413581878413672875874945, −17.21342329948936360352243998847, −16.524137319672460351062309579697, −15.70223119694898231287486558054, −14.4335809616532921093380684343, −14.17390452256991688710505356296, −13.127267733328488121054410919442, −12.1643008575343539425538082870, −11.596682480133975591061814688925, −10.544302076660340996480919441190, −9.50143409058250169210691931233, −8.82196679585560463454614906730, −8.241145206044028066394780732219, −6.78185400693291132044430598471, −6.111301911613901173635878738726, −5.03998934553592224275086811771, −4.50739944987386699974715518371, −2.94200437413414881445193359294, −2.06670043174914758860637371909, −1.019934904956319408514891235165, 1.241798646962436463013459026460, 2.10287295964494506541349819083, 3.280493485375695550762070341, 4.32065904529571684649683750306, 5.2819599183650995537646964475, 6.26164184290252584745837845743, 7.12644259887061896639714407821, 7.89546168946700421349341378332, 8.97388018348439539395587658291, 10.18335283071004865394400633873, 10.34643587620664932909720623324, 11.40774666652082655843510201654, 12.506627507635028874460393977905, 13.21271044949170667270840770759, 14.25441210857672827723548974471, 14.68931079227131373566828330068, 15.466969600182084351731550690903, 16.89029629939419711555310855192, 17.46929322312155543905353436015, 17.73658573159023888813763781372, 19.033378086516510769585334652800, 19.65993996969256478654740346392, 20.69685420497341021010771237700, 21.21740505951233290065796214155, 22.09863545328069613785989014123

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