↑

Properties

Label 2-2e8-8.5-c7-0-46
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  − 90.5i·3-s + 482. i·5-s + 644.·7-s − 6.00e3·9-s − 2.59e3i·11-s + 2.54e3i·13-s + 4.36e4·15-s + 1.55e4·17-s − 2.51e4i·19-s − 5.83e4i·21-s + 1.81e4·23-s − 1.54e5·25-s + 3.46e5i·27-s + 2.89e4i·29-s − 4.89e4·31-s + ⋯
L(s)  = 1  − 1.93i·3-s + 1.72i·5-s + 0.710·7-s − 2.74·9-s − 0.587i·11-s + 0.320i·13-s + 3.34·15-s + 0.768·17-s − 0.842i·19-s − 1.37i·21-s + 0.311·23-s − 1.98·25-s + 3.38i·27-s + 0.220i·29-s − 0.294·31-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

\(L(4)\) \(\approx\) \(0.2248410869\)
\(L(\frac12)\) \(\approx\) \(0.2248410869\)
\(L(\frac{9}{2})\) 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 \( 1 + 90.5iT - 2.18e3T^{2} \)
5 \( 1 - 482. iT - 7.81e4T^{2} \)
7 \( 1 - 644.T + 8.23e5T^{2} \)
11 \( 1 + 2.59e3iT - 1.94e7T^{2} \)
13 \( 1 - 2.54e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.55e4T + 4.10e8T^{2} \)
19 \( 1 + 2.51e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.81e4T + 3.40e9T^{2} \)
29 \( 1 - 2.89e4iT - 1.72e10T^{2} \)
31 \( 1 + 4.89e4T + 2.75e10T^{2} \)
37 \( 1 + 3.68e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.36e5T + 1.94e11T^{2} \)
43 \( 1 + 6.06e5iT - 2.71e11T^{2} \)
47 \( 1 + 8.69e5T + 5.06e11T^{2} \)
53 \( 1 - 1.29e6iT - 1.17e12T^{2} \)
59 \( 1 + 4.32e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.87e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.79e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.53e6T + 9.09e12T^{2} \)
73 \( 1 + 1.24e6T + 1.10e13T^{2} \)
79 \( 1 + 2.83e6T + 1.92e13T^{2} \)
83 \( 1 - 1.91e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.94e6T + 4.42e13T^{2} \)
97 \( 1 + 7.26e6T + 8.07e13T^{2} \)
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

−10.58333969586490165489529826914, −8.877988669728302880465930644853, −7.80598205539549135634311719800, −7.18357263813511716009437879112, −6.47834609734013079974315441154, −5.52818955384147144983526934543, −3.29678552132669525478624616399, −2.44454253289088751552686473603, −1.42521814779693819839709081215, −0.05026860452200809230653491820, 1.47983865734639968901854024272, 3.35032277134834325003662603871, 4.49960073863525759302543958600, 4.94744883349478038241822430265, 5.75815319746496956079904831529, 8.091916135941490299188205881162, 8.552262056916693800367687868094, 9.662303510368512205031216133381, 10.04572338756967503319543263635, 11.33488774552170561866366361653

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