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Properties

Label 1-2e8-256.5-r0-0-0
Degree $1$
Conductor $256$
Sign $0.689 + 0.724i$
Analytic cond. $1.18885$
Root an. cond. $1.18885$
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  + (−0.956 − 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.555 + 0.831i)7-s + (0.831 + 0.555i)9-s + (−0.471 + 0.881i)11-s + (−0.0980 − 0.995i)13-s + (−0.923 − 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.634 + 0.773i)19-s + (−0.290 − 0.956i)21-s + (0.195 + 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.634 − 0.773i)27-s + (0.881 − 0.471i)29-s + (0.707 + 0.707i)31-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.555 + 0.831i)7-s + (0.831 + 0.555i)9-s + (−0.471 + 0.881i)11-s + (−0.0980 − 0.995i)13-s + (−0.923 − 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.634 + 0.773i)19-s + (−0.290 − 0.956i)21-s + (0.195 + 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.634 − 0.773i)27-s + (0.881 − 0.471i)29-s + (0.707 + 0.707i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(1.18885\)
Root analytic conductor: \(1.18885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 256,\ (0:\ ),\ 0.689 + 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9552025583 + 0.4094620094i\)
\(L(\frac12)\) \(\approx\) \(0.9552025583 + 0.4094620094i\)
\(L(1)\) \(\approx\) \(0.9420068876 + 0.1416823426i\)
\(L(1)\) \(\approx\) \(0.9420068876 + 0.1416823426i\)

Euler product

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

−26.239688792746459848726062830441, −24.49572384383513737640519904535, −24.13762820125150212069772032380, −23.086276456024946275299417559238, −22.05816449092630634114505450508, −21.28666949669695875293402627067, −20.72943142034014331512254634143, −19.255040321756112883504813759201, −18.094239924311037903987782586339, −17.44099167001933969774644197393, −16.66295370189029488314197188560, −15.85203983644468255437306812754, −14.39431901071719587571856722257, −13.59050024998275493853991505791, −12.62933843554648715695554706983, −11.17094944156803592767684880216, −10.79909963220903963345035567681, −9.65656642017315295059071115214, −8.59489489984126814402803360624, −6.953981521341054977389688128456, −6.25004948569917023859445668033, −4.992699331264459675290009415744, −4.28040117878885180939483530327, −2.38531450671633656405705947288, −0.875938330710494984045223933642, 1.55682942159590248026551686060, 2.51130335790048532569327066619, 4.616826094745637880685455145176, 5.4966873605428192246453643824, 6.25041963387710757107345266893, 7.475219827155962147986541716272, 8.70380266297080032176885930611, 10.08871209244743228256040509162, 10.66254043799978517598415589570, 11.97134070569330930242284417501, 12.719335231573504458821691794049, 13.61651366582648542331203356121, 15.00446456149156147832827152372, 15.71713863024185642943087821645, 17.19936953299672004292445743250, 17.712383956949598379132888884219, 18.26205705128385435157434739030, 19.4572006323947527896960127106, 20.926494520520058982369899307520, 21.49493402946753255743002038160, 22.467610522024448825517168025334, 23.17236482130191536079731071440, 24.392448529377859314047633563119, 25.01294389783185697713703240363, 25.82722462444359779394429371818

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