↑

Properties

Label 2-2e8-8.5-c3-0-13
Degree $2$
Conductor $256$
Sign $0.707 + 0.707i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·3-s − 2i·5-s − 24·7-s + 11·9-s − 44i·11-s − 22i·13-s + 8·15-s + 50·17-s − 44i·19-s − 96i·21-s + 56·23-s + 121·25-s + 152i·27-s − 198i·29-s − 160·31-s + ⋯
L(s)  = 1  + 0.769i·3-s − 0.178i·5-s − 1.29·7-s + 0.407·9-s − 1.20i·11-s − 0.469i·13-s + 0.137·15-s + 0.713·17-s − 0.531i·19-s − 0.997i·21-s + 0.507·23-s + 0.967·25-s + 1.08i·27-s − 1.26i·29-s − 0.926·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.380062326\)
\(L(\frac12)\) \(\approx\) \(1.380062326\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 4iT - 27T^{2} \)
5 \( 1 + 2iT - 125T^{2} \)
7 \( 1 + 24T + 343T^{2} \)
11 \( 1 + 44iT - 1.33e3T^{2} \)
13 \( 1 + 22iT - 2.19e3T^{2} \)
17 \( 1 - 50T + 4.91e3T^{2} \)
19 \( 1 + 44iT - 6.85e3T^{2} \)
23 \( 1 - 56T + 1.21e4T^{2} \)
29 \( 1 + 198iT - 2.43e4T^{2} \)
31 \( 1 + 160T + 2.97e4T^{2} \)
37 \( 1 + 162iT - 5.06e4T^{2} \)
41 \( 1 - 198T + 6.89e4T^{2} \)
43 \( 1 - 52iT - 7.95e4T^{2} \)
47 \( 1 - 528T + 1.03e5T^{2} \)
53 \( 1 + 242iT - 1.48e5T^{2} \)
59 \( 1 + 668iT - 2.05e5T^{2} \)
61 \( 1 + 550iT - 2.26e5T^{2} \)
67 \( 1 + 188iT - 3.00e5T^{2} \)
71 \( 1 + 728T + 3.57e5T^{2} \)
73 \( 1 + 154T + 3.89e5T^{2} \)
79 \( 1 + 656T + 4.93e5T^{2} \)
83 \( 1 + 236iT - 5.71e5T^{2} \)
89 \( 1 + 714T + 7.04e5T^{2} \)
97 \( 1 + 478T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.22764714299319912895341675107, −10.42538900663903027115445844205, −9.550780307567013017507173911884, −8.847881655349570779554331129719, −7.48749552587012996451964213837, −6.28950060603509690117392218994, −5.28002614422809057896061428657, −3.87861137604062335193575549335, −2.98632984558379587973162317561, −0.59803423341275942997376131769, 1.35557532313524436370529560537, 2.86153153938665675655990324674, 4.26080451446738959082772263379, 5.81526475236632263694665973477, 7.05410609786882479615949649298, 7.24457423617757938796820382894, 8.886174068167655909965859509063, 9.820017050472189887352634511801, 10.57723738449281527308274480460, 12.11612707463161875523685853223

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