↑

Properties

Label 2-2e8-8.5-c7-0-37
Degree $2$
Conductor $256$
Sign $0.707 + 0.707i$
Analytic cond. $79.9705$
Root an. cond. $8.94262$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 84i·3-s − 82i·5-s + 456·7-s − 4.86e3·9-s − 2.52e3i·11-s + 1.07e4i·13-s + 6.88e3·15-s − 1.11e4·17-s − 4.12e3i·19-s + 3.83e4i·21-s − 8.17e4·23-s + 7.14e4·25-s − 2.25e5i·27-s − 9.97e4i·29-s − 4.04e4·31-s + ⋯
L(s)  = 1  + 1.79i·3-s − 0.293i·5-s + 0.502·7-s − 2.22·9-s − 0.571i·11-s + 1.36i·13-s + 0.526·15-s − 0.550·17-s − 0.137i·19-s + 0.902i·21-s − 1.40·23-s + 0.913·25-s − 2.20i·27-s − 0.759i·29-s − 0.244·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(79.9705\)
Root analytic conductor: \(8.94262\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :7/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6700756648\)
\(L(\frac12)\) \(\approx\) \(0.6700756648\)
\(L(\frac{9}{2})\) 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 \( 1 - 84iT - 2.18e3T^{2} \)
5 \( 1 + 82iT - 7.81e4T^{2} \)
7 \( 1 - 456T + 8.23e5T^{2} \)
11 \( 1 + 2.52e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.07e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.11e4T + 4.10e8T^{2} \)
19 \( 1 + 4.12e3iT - 8.93e8T^{2} \)
23 \( 1 + 8.17e4T + 3.40e9T^{2} \)
29 \( 1 + 9.97e4iT - 1.72e10T^{2} \)
31 \( 1 + 4.04e4T + 2.75e10T^{2} \)
37 \( 1 + 4.19e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.41e5T + 1.94e11T^{2} \)
43 \( 1 + 6.90e5iT - 2.71e11T^{2} \)
47 \( 1 + 6.82e5T + 5.06e11T^{2} \)
53 \( 1 - 1.81e6iT - 1.17e12T^{2} \)
59 \( 1 + 9.66e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.88e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.96e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.54e6T + 9.09e12T^{2} \)
73 \( 1 - 1.68e6T + 1.10e13T^{2} \)
79 \( 1 - 4.03e6T + 1.92e13T^{2} \)
83 \( 1 - 5.38e6iT - 2.71e13T^{2} \)
89 \( 1 - 6.47e6T + 4.42e13T^{2} \)
97 \( 1 + 6.06e6T + 8.07e13T^{2} \)
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

−10.72532125753540279067780307607, −9.577656471168676227115989819965, −9.004117622771594297649082735016, −8.105087292105323844963876075342, −6.40310702394613329366114798368, −5.24091705598414179955281739247, −4.40262392439682416425100183962, −3.64416350855205688470013100548, −2.12233151472190567563808185408, −0.16054144682128735812978866171, 1.09218341792945546554962518828, 2.05363110490017659800337975830, 3.12577401834294823773825107298, 5.01206895375848133957518027066, 6.17195982811732043725960572218, 6.98445456768812272635542563173, 7.921895020352833980086120407616, 8.453648364449749936029808142988, 10.04218243679370980611432845593, 11.14925570086343818736714994976

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