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Properties

Label 1-459-459.250-r1-0-0
Degree $1$
Conductor $459$
Sign $-0.701 - 0.712i$
Analytic cond. $49.3264$
Root an. cond. $49.3264$
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  + (−0.0871 + 0.996i)2-s + (−0.984 − 0.173i)4-s + (−0.999 − 0.0436i)5-s + (0.537 + 0.843i)7-s + (0.258 − 0.965i)8-s + (0.130 − 0.991i)10-s + (0.0436 + 0.999i)11-s + (−0.642 + 0.766i)13-s + (−0.887 + 0.461i)14-s + (0.939 + 0.342i)16-s + (0.965 + 0.258i)19-s + (0.976 + 0.216i)20-s + (−0.999 − 0.0436i)22-s + (0.976 − 0.216i)23-s + (0.996 + 0.0871i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.0871 + 0.996i)2-s + (−0.984 − 0.173i)4-s + (−0.999 − 0.0436i)5-s + (0.537 + 0.843i)7-s + (0.258 − 0.965i)8-s + (0.130 − 0.991i)10-s + (0.0436 + 0.999i)11-s + (−0.642 + 0.766i)13-s + (−0.887 + 0.461i)14-s + (0.939 + 0.342i)16-s + (0.965 + 0.258i)19-s + (0.976 + 0.216i)20-s + (−0.999 − 0.0436i)22-s + (0.976 − 0.216i)23-s + (0.996 + 0.0871i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(49.3264\)
Root analytic conductor: \(49.3264\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 459,\ (1:\ ),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3526297113 + 0.8420359978i\)
\(L(\frac12)\) \(\approx\) \(-0.3526297113 + 0.8420359978i\)
\(L(1)\) \(\approx\) \(0.5529350035 + 0.5601918748i\)
\(L(1)\) \(\approx\) \(0.5529350035 + 0.5601918748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.0871 + 0.996i)T \)
5 \( 1 + (-0.999 - 0.0436i)T \)
7 \( 1 + (0.537 + 0.843i)T \)
11 \( 1 + (0.0436 + 0.999i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.976 - 0.216i)T \)
29 \( 1 + (-0.953 + 0.300i)T \)
31 \( 1 + (0.843 + 0.537i)T \)
37 \( 1 + (0.608 + 0.793i)T \)
41 \( 1 + (0.953 + 0.300i)T \)
43 \( 1 + (0.422 + 0.906i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.906 - 0.422i)T \)
61 \( 1 + (-0.537 - 0.843i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.608 + 0.793i)T \)
73 \( 1 + (-0.130 - 0.991i)T \)
79 \( 1 + (-0.887 - 0.461i)T \)
83 \( 1 + (0.0871 - 0.996i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.737 + 0.675i)T \)
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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

−22.90127712791395448386870143217, −22.45972128591992606038022426716, −21.27275978737056170871096661002, −20.53432199505191425755927996883, −19.74255257389243496127839854374, −19.18441272400895891316998166324, −18.206199047233060362073888487546, −17.27384857745514577320719307490, −16.47181080545672613506793610549, −15.23416840251294216853748177686, −14.30267993581524565134342909021, −13.42938704705651053924859676495, −12.53954712810847112675088892327, −11.39083922457654103241642381219, −11.121460727702675549573340595398, −10.06612846132036070641220968745, −8.94968883442833109673011273589, −7.92310834226212012384997349969, −7.360833599412636279387240923265, −5.51970488069663531172657279058, −4.52608830223940746879206018761, −3.60948693583698044835197449527, −2.766680407197620322867285012962, −1.069598814737074439524031836240, −0.31554627282345216817278729822, 1.383723852747531950926167446444, 3.06550958478335921311735361262, 4.55819430505834916209362640409, 4.89822476129989799901071562670, 6.287243300054372346928771574306, 7.350123952121952120974119272895, 7.89935493252611356067424130938, 9.00765873834218348629396011669, 9.67418272057797779628532306219, 11.161138920298188720184183309775, 12.11230922533970729498004542314, 12.81241970557879643092174364114, 14.251247044676088537048462176343, 14.86396319563550752257697069050, 15.52197947372370691773392533513, 16.37290144353455709383988815663, 17.27450102688013442664863635435, 18.2278418154719743022096265139, 18.8885467373573896119671699010, 19.77908349406661220377971956461, 20.90238348909570714719464111390, 21.98715450712832195610856927852, 22.7795918764381114655883729097, 23.4549266212719054035169644554, 24.50354567970616820668127176694

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