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Properties

Label 1-864-864.587-r0-0-0
Degree $1$
Conductor $864$
Sign $-0.614 + 0.788i$
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  + (−0.906 − 0.422i)5-s + (0.984 − 0.173i)7-s + (−0.422 − 0.906i)11-s + (−0.996 + 0.0871i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.0871 + 0.996i)29-s + (−0.173 + 0.984i)31-s + (−0.965 − 0.258i)35-s + (0.965 − 0.258i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)47-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)5-s + (0.984 − 0.173i)7-s + (−0.422 − 0.906i)11-s + (−0.996 + 0.0871i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.0871 + 0.996i)29-s + (−0.173 + 0.984i)31-s + (−0.965 − 0.258i)35-s + (0.965 − 0.258i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.614 + 0.788i$
Analytic conductor: \(4.01239\)
Root analytic conductor: \(4.01239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 864,\ (0:\ ),\ -0.614 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1320009651 + 0.2701433020i\)
\(L(\frac12)\) \(\approx\) \(0.1320009651 + 0.2701433020i\)
\(L(1)\) \(\approx\) \(0.7294501682 + 0.02148858072i\)
\(L(1)\) \(\approx\) \(0.7294501682 + 0.02148858072i\)

Euler product

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

−21.99673113642404925082688089989, −20.700486692664127908423937026079, −20.36800110577973658449930818367, −19.32731745736499178488117592310, −18.62901074207761539305448751944, −17.781397704489838442190173570627, −17.16007468330988659165178921021, −15.98963177073649513668461295482, −15.18810835685547879008314427063, −14.772845406511032220378144578352, −13.85107011134256351017187602847, −12.6657669797510035512803132453, −11.92374094134003574576537821115, −11.30198228804843539940047113564, −10.375240579329266301242636079291, −9.51119972935170722525505524577, −8.28889798738802044651019305082, −7.67148084587445560589542889062, −7.018622099196524903067606632430, −5.75298079853687500100464449749, −4.580790960214541315847625008114, −4.20518892903991381041131052955, −2.65852107016553520724296027663, −1.99618232237954981434094322995, −0.133095486206310456636165175787, 1.341650379759235502313703545098, 2.52103152199571073649226862391, 3.78158244667108899166318503959, 4.56969980159849007508837852008, 5.32936741598133173533247726326, 6.5580417650577714414532386866, 7.60751813330421666960383073598, 8.304107404965350526694524147301, 8.84503525103395754982197376778, 10.2692835645856762586326866023, 11.00569929279072267032644271160, 11.70914741545041089684047478876, 12.58448387311550615496064075303, 13.35360045263606395369372981278, 14.60455737824648985489439107752, 14.893961601480297345928803040757, 16.07543574291766236082334981200, 16.629875128708287460332547438148, 17.56759741249416905526637678080, 18.31590975821429877315963386774, 19.45931484635252642214518238512, 19.746315781504682724742452758129, 20.75719078150343557927813962895, 21.57054780482564057834100360467, 22.14055383384295021339278597933

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