Properties

Label 1-2e6-64.5-r0-0-0
Degree $1$
Conductor $64$
Sign $0.634 + 0.773i$
Analytic cond. $0.297214$
Root an. cond. $0.297214$
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  + (0.382 + 0.923i)3-s + (0.923 + 0.382i)5-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)13-s + i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.382 − 0.923i)29-s − 31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.923 + 0.382i)5-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)13-s + i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.382 − 0.923i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(0.297214\)
Root analytic conductor: \(0.297214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (0:\ ),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9586907000 + 0.4534269321i\)
\(L(\frac12)\) \(\approx\) \(0.9586907000 + 0.4534269321i\)
\(L(1)\) \(\approx\) \(1.103319112 + 0.3385797248i\)
\(L(1)\) \(\approx\) \(1.103319112 + 0.3385797248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 - T \)
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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

−32.011096407705070349233394780967, −31.0849621364813562537218827219, −29.72266435273047508877109248813, −29.03937499531797899884259282473, −28.05003098386480199185992844359, −25.98892277352743353397159398426, −25.60836958179665724073823456782, −24.39827437036053570347049168176, −23.48535085486100316111736550306, −21.85502846710737684552901093400, −20.94471290841906939370173284783, −19.42326223172920093413802511480, −18.62530360519801950130414946241, −17.469052900381844741388388931726, −16.164226387982578871708495214372, −14.58301012384432715650466098400, −13.237023294488903619044807616737, −12.78677290303939034347855293581, −11.04589942041792045045910025292, −9.21882219366086194265759404554, −8.4464489738136899646827164789, −6.52943091678980316039062948736, −5.6965377329053514766056664882, −3.189313832646864378474855163316, −1.688359758410706679331512679678, 2.498428486412202240768336624240, 3.96910493400335712751711719044, 5.56690554871849515191828521469, 7.11813004944074725412207890767, 8.953443959306035956208098964620, 10.0724887365345354937158277139, 10.78633022486154205693127908843, 12.96796119952299669664313030538, 13.97358783883417042832449756821, 15.166580337377798463414526125530, 16.36595331507539501380790301153, 17.470298579754257426432358037862, 18.878157943525390729449708229648, 20.42969700937700515173475737949, 20.96518881824040102334049177304, 22.44471636748367210645202510909, 23.06677202563200077524712833237, 25.18654165990104125698088403989, 25.796158802108026199878894031294, 26.70730903401464501529457082337, 28.003298046897955936493752197394, 29.08952000091600276096268261521, 30.2236644165998725098572944233, 31.46357620867303411548691054836, 32.73215844500233758450981100895

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