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Properties

Label 1-2e8-256.147-r1-0-0
Degree $1$
Conductor $256$
Sign $0.0245 + 0.999i$
Analytic cond. $27.5110$
Root an. cond. $27.5110$
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  + (−0.881 − 0.471i)3-s + (0.634 − 0.773i)5-s + (0.831 + 0.555i)7-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (−0.773 + 0.634i)13-s + (−0.923 + 0.382i)15-s + (0.923 + 0.382i)17-s + (−0.0980 + 0.995i)19-s + (−0.471 − 0.881i)21-s + (−0.980 − 0.195i)23-s + (−0.195 − 0.980i)25-s + (−0.0980 − 0.995i)27-s + (−0.290 − 0.956i)29-s + (−0.707 + 0.707i)31-s + ⋯
L(s)  = 1  + (−0.881 − 0.471i)3-s + (0.634 − 0.773i)5-s + (0.831 + 0.555i)7-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (−0.773 + 0.634i)13-s + (−0.923 + 0.382i)15-s + (0.923 + 0.382i)17-s + (−0.0980 + 0.995i)19-s + (−0.471 − 0.881i)21-s + (−0.980 − 0.195i)23-s + (−0.195 − 0.980i)25-s + (−0.0980 − 0.995i)27-s + (−0.290 − 0.956i)29-s + (−0.707 + 0.707i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.0245 + 0.999i$
Analytic conductor: \(27.5110\)
Root analytic conductor: \(27.5110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 256,\ (1:\ ),\ 0.0245 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5976205397 + 0.5831298253i\)
\(L(\frac12)\) \(\approx\) \(0.5976205397 + 0.5831298253i\)
\(L(1)\) \(\approx\) \(0.8133832279 + 0.01966050703i\)
\(L(1)\) \(\approx\) \(0.8133832279 + 0.01966050703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.881 - 0.471i)T \)
5 \( 1 + (0.634 - 0.773i)T \)
7 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (-0.956 - 0.290i)T \)
13 \( 1 + (-0.773 + 0.634i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (-0.0980 + 0.995i)T \)
23 \( 1 + (-0.980 - 0.195i)T \)
29 \( 1 + (-0.290 - 0.956i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.995 + 0.0980i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (0.881 - 0.471i)T \)
47 \( 1 + (-0.382 + 0.923i)T \)
53 \( 1 + (-0.290 + 0.956i)T \)
59 \( 1 + (0.773 + 0.634i)T \)
61 \( 1 + (-0.471 + 0.881i)T \)
67 \( 1 + (-0.471 + 0.881i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (-0.831 + 0.555i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.995 + 0.0980i)T \)
89 \( 1 + (0.980 - 0.195i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−25.78030618080811323126913113036, −24.33375289886098227353284696283, −23.60319868844015955642737969734, −22.65846958608857773683942909480, −21.89798195017095943203419925949, −21.06665207093434086407782518274, −20.25201919585139195460427349317, −18.69658975756045402620828746860, −17.78046778894435850857455106831, −17.428978414606211803095972771330, −16.2312465709841406863301581571, −15.14324371239549292137615713032, −14.37990156940347054367958268138, −13.20295840542924457404091631957, −12.04842596256332744491494289067, −10.914018612688676693728933838423, −10.38068889006534276460275751806, −9.507870175264389426989901028018, −7.72860655915351437930916905862, −6.94794039229830903386916058238, −5.52037587893493452722829327087, −4.973498003652428816634193818213, −3.46745102000370741842382287091, −2.010205245917131515196980736447, −0.29355399490606552974208922118, 1.358555058501406872377902786524, 2.25906121628065759578711309619, 4.44293662536545044838782877900, 5.43166541057571776801452671577, 5.97377908186584358242188306505, 7.580859569877836447510305489309, 8.3684028585115472288663887875, 9.77905742701009211783497878799, 10.692721088572617204257761321530, 12.02095660006406961272355368702, 12.407014629233575913669607039382, 13.59546127435617472459315039042, 14.54427166591135196123815127958, 16.00151708249122093130537639313, 16.7452749473880171169098721805, 17.59759139158168327666978882801, 18.387489090661692267028271965721, 19.23005934363702057184380936289, 20.77530331093200933939681997846, 21.34657666142153667991564318694, 22.175559782696331928046944307157, 23.47596822062330344283020398883, 24.137888700375827449722404307698, 24.74476354252627461278629500433, 25.745532359449097001018374046791

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