Properties

Degree $2$
Conductor $256$
Sign $0.707 - 0.707i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·3-s − 14i·5-s − 208·7-s + 179·9-s − 536i·11-s + 694i·13-s + 112·15-s − 1.27e3·17-s − 1.11e3i·19-s − 1.66e3i·21-s + 3.21e3·23-s + 2.92e3·25-s + 3.37e3i·27-s + 2.91e3i·29-s + 2.62e3·31-s + ⋯
L(s)  = 1  + 0.513i·3-s − 0.250i·5-s − 1.60·7-s + 0.736·9-s − 1.33i·11-s + 1.13i·13-s + 0.128·15-s − 1.07·17-s − 0.706i·19-s − 0.823i·21-s + 1.26·23-s + 0.937·25-s + 0.891i·27-s + 0.644i·29-s + 0.490·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 - 0.707i$
Motivic weight: \(5\)
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.532736939\)
\(L(\frac12)\) \(\approx\) \(1.532736939\)
\(L(\frac{7}{2})\) 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 \( 1 - 8iT - 243T^{2} \)
5 \( 1 + 14iT - 3.12e3T^{2} \)
7 \( 1 + 208T + 1.68e4T^{2} \)
11 \( 1 + 536iT - 1.61e5T^{2} \)
13 \( 1 - 694iT - 3.71e5T^{2} \)
17 \( 1 + 1.27e3T + 1.41e6T^{2} \)
19 \( 1 + 1.11e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.21e3T + 6.43e6T^{2} \)
29 \( 1 - 2.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.62e3T + 2.86e7T^{2} \)
37 \( 1 - 9.45e3iT - 6.93e7T^{2} \)
41 \( 1 + 170T + 1.15e8T^{2} \)
43 \( 1 + 1.99e4iT - 1.47e8T^{2} \)
47 \( 1 + 32T + 2.29e8T^{2} \)
53 \( 1 - 2.21e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.14e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.54e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.07e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.85e4T + 1.80e9T^{2} \)
73 \( 1 - 5.36e4T + 2.07e9T^{2} \)
79 \( 1 - 6.91e4T + 3.07e9T^{2} \)
83 \( 1 - 3.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.26e5T + 5.58e9T^{2} \)
97 \( 1 - 6.22e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.09851209440181866137820947320, −10.33429823638872937654033202665, −9.069069964767140883089591752317, −8.963908952979392172757969069897, −6.97465308288673453044153094673, −6.46851200471853872958380150271, −4.99245455712711890759791353660, −3.86253088283351843367926703351, −2.81658386010271444146530465837, −0.853137605783157883804759930761, 0.60220164904891101494303350877, 2.24198800218706059853707855321, 3.43552373123927133286410015725, 4.79850606583301780160859740845, 6.36974902341444197604499602974, 6.90115224621128176052192026829, 7.88130016279287376779156749749, 9.371347617413296120511070335792, 9.976498169093567598215292574664, 10.89628636939291445074987756995

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