Properties

Label 2-2e8-4.3-c0-0-0
Degree $2$
Conductor $256$
Sign $1$
Analytic cond. $0.127760$
Root an. cond. $0.357436$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 2·17-s − 25-s − 2·41-s + 49-s + 2·73-s + 81-s + 2·89-s − 2·97-s + 2·113-s + ⋯
L(s)  = 1  + 9-s − 2·17-s − 25-s − 2·41-s + 49-s + 2·73-s + 81-s + 2·89-s − 2·97-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(0.127760\)
Root analytic conductor: \(0.357436\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{256} (255, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7559661027\)
\(L(\frac12)\) \(\approx\) \(0.7559661027\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 + T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 + T )^{2} \)
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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

−12.26505903510516084641264694255, −11.27425088452062960443173901726, −10.35217709081203788520341877720, −9.411931474298749010055050808183, −8.412844904341925196918485700193, −7.21695810024518924745425559826, −6.38107797641790162398645317460, −4.90430441146662750205657053043, −3.86766998712323449552352878936, −2.05854479506742573974032235914, 2.05854479506742573974032235914, 3.86766998712323449552352878936, 4.90430441146662750205657053043, 6.38107797641790162398645317460, 7.21695810024518924745425559826, 8.412844904341925196918485700193, 9.411931474298749010055050808183, 10.35217709081203788520341877720, 11.27425088452062960443173901726, 12.26505903510516084641264694255

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