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Properties

Label 1-864-864.637-r0-0-0
Degree $1$
Conductor $864$
Sign $0.265 - 0.964i$
Analytic cond. $4.01239$
Root an. cond. $4.01239$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.996 + 0.0871i)5-s + (−0.342 + 0.939i)7-s + (0.0871 − 0.996i)11-s + (−0.819 + 0.573i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.573 + 0.819i)29-s + (−0.939 + 0.342i)31-s + (0.258 − 0.965i)35-s + (−0.258 − 0.965i)37-s + (0.984 + 0.173i)41-s + (0.0871 − 0.996i)43-s + (0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)5-s + (−0.342 + 0.939i)7-s + (0.0871 − 0.996i)11-s + (−0.819 + 0.573i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.573 + 0.819i)29-s + (−0.939 + 0.342i)31-s + (0.258 − 0.965i)35-s + (−0.258 − 0.965i)37-s + (0.984 + 0.173i)41-s + (0.0871 − 0.996i)43-s + (0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(4.01239\)
Root analytic conductor: \(4.01239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 864,\ (0:\ ),\ 0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5375929040 - 0.4094124937i\)
\(L(\frac12)\) \(\approx\) \(0.5375929040 - 0.4094124937i\)
\(L(1)\) \(\approx\) \(0.7523278322 + 0.02702474413i\)
\(L(1)\) \(\approx\) \(0.7523278322 + 0.02702474413i\)

Euler product

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

−22.459775270133627235294988324, −21.418580732640177018168111350258, −20.33993800792542239309457669540, −19.80662008971960179213657345171, −19.37380935251782570732687640031, −18.241058123675686012634419960354, −17.15737112143960643128305813614, −16.88495059661405663519423034514, −15.58012598176134893660322990103, −15.191065505960569015560480996999, −14.28151442049181296146986392633, −13.118973476952276452023054111466, −12.57815233543824682932526680589, −11.67609002540914514118342344887, −10.773888053391480311041729421159, −9.980025374439723336427810851, −9.14158410749954413905795214957, −7.69878448996423947894336798924, −7.590285695689621954169829146014, −6.53676329321831376496693250035, −5.24604740760417681955932148051, −4.26328524159653305148985078084, −3.67227583720611155895191746293, −2.463885439810915713426542422967, −1.03365246762784861872297908667, 0.35688651092379325175540153437, 2.064649796274279108800403751492, 3.10798104149567718791837199924, 3.8916929002377429453106119020, 5.05929225994561405126475395176, 5.91269579064354011172551679469, 6.96368606205345599040965449291, 7.79695258406642953367214945610, 8.73629688621805459987216582814, 9.358318032473182344592157142993, 10.582704855556865170898297624101, 11.38119409186033374632717282226, 12.28498230243844501368812049401, 12.5692008340287959382703379259, 14.135687502075645820640935550232, 14.55132853827126882960444598287, 15.58136592113209974168913608647, 16.3374991024479125993525069324, 16.737175601341908051658740771815, 18.30387192929637845384875437241, 18.71421066999741078365499623126, 19.37962727074699095769574505375, 20.19682931085029947164102550232, 21.17822434407260311486211809935, 21.97864477415717979652994325284

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