↑

Properties

Label 2-2e8-8.5-c3-0-6
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  + 8i·3-s − 10i·5-s + 16·7-s − 37·9-s + 40i·11-s + 50i·13-s + 80·15-s − 30·17-s + 40i·19-s + 128i·21-s + 48·23-s + 25·25-s − 80i·27-s + 34i·29-s − 320·31-s + ⋯
L(s)  = 1  + 1.53i·3-s − 0.894i·5-s + 0.863·7-s − 1.37·9-s + 1.09i·11-s + 1.06i·13-s + 1.37·15-s − 0.428·17-s + 0.482i·19-s + 1.33i·21-s + 0.435·23-s + 0.200·25-s − 0.570i·27-s + 0.217i·29-s − 1.85·31-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

\(L(2)\) \(\approx\) \(1.597575222\)
\(L(\frac12)\) \(\approx\) \(1.597575222\)
\(L(\frac{5}{2})\) 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 \( 1 - 8iT - 27T^{2} \)
5 \( 1 + 10iT - 125T^{2} \)
7 \( 1 - 16T + 343T^{2} \)
11 \( 1 - 40iT - 1.33e3T^{2} \)
13 \( 1 - 50iT - 2.19e3T^{2} \)
17 \( 1 + 30T + 4.91e3T^{2} \)
19 \( 1 - 40iT - 6.85e3T^{2} \)
23 \( 1 - 48T + 1.21e4T^{2} \)
29 \( 1 - 34iT - 2.43e4T^{2} \)
31 \( 1 + 320T + 2.97e4T^{2} \)
37 \( 1 - 310iT - 5.06e4T^{2} \)
41 \( 1 + 410T + 6.89e4T^{2} \)
43 \( 1 + 152iT - 7.95e4T^{2} \)
47 \( 1 - 416T + 1.03e5T^{2} \)
53 \( 1 + 410iT - 1.48e5T^{2} \)
59 \( 1 - 200iT - 2.05e5T^{2} \)
61 \( 1 + 30iT - 2.26e5T^{2} \)
67 \( 1 - 776iT - 3.00e5T^{2} \)
71 \( 1 - 400T + 3.57e5T^{2} \)
73 \( 1 - 630T + 3.89e5T^{2} \)
79 \( 1 - 1.12e3T + 4.93e5T^{2} \)
83 \( 1 - 552iT - 5.71e5T^{2} \)
89 \( 1 - 326T + 7.04e5T^{2} \)
97 \( 1 + 110T + 9.12e5T^{2} \)
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

−11.74270911886240281870207657876, −10.90066341406279168920037281994, −9.922421216994580345763145502540, −9.132834781360296760594847863853, −8.443703117772995318659693451716, −7.00336017404331775659543717528, −5.22832895826117415389890931193, −4.72413864921906530323589817852, −3.82277485796557956693962682318, −1.79119022451592537989005078492, 0.63301741996652784432489114438, 2.10056678164313028491213512821, 3.28816367749648520267647967832, 5.33395294850062952231818612913, 6.35509400858744123006377628469, 7.30140909751366380493270886787, 7.987788004469972135605515317038, 8.967081238669298308113551414543, 10.82786542846412693262239802028, 11.07188218745043156842696488615

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