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Properties

Label 2-2e8-1.1-c7-0-28
Degree $2$
Conductor $256$
Sign $-1$
Analytic cond. $79.9705$
Root an. cond. $8.94262$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 556·5-s − 2.18e3·9-s + 1.31e4·13-s + 4.00e4·17-s + 2.31e5·25-s − 1.20e5·29-s − 2.48e5·37-s − 9.53e3·41-s + 1.21e6·45-s − 8.23e5·49-s − 2.01e6·53-s − 5.32e5·61-s − 7.28e6·65-s + 3.91e6·73-s + 4.78e6·81-s − 2.22e7·85-s + 9.24e6·89-s + 1.75e7·97-s + 1.58e7·101-s − 1.22e7·109-s − 2.91e7·113-s − 2.86e7·117-s + ⋯
L(s)  = 1  − 1.98·5-s − 9-s + 1.65·13-s + 1.97·17-s + 2.95·25-s − 0.920·29-s − 0.805·37-s − 0.0215·41-s + 1.98·45-s − 49-s − 1.85·53-s − 0.300·61-s − 3.29·65-s + 1.17·73-s + 81-s − 3.93·85-s + 1.39·89-s + 1.95·97-s + 1.53·101-s − 0.904·109-s − 1.89·113-s − 1.65·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-1$
Analytic conductor: \(79.9705\)
Root analytic conductor: \(8.94262\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 256,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + p^{7} T^{2} \)
5 \( 1 + 556 T + p^{7} T^{2} \)
7 \( 1 + p^{7} T^{2} \)
11 \( 1 + p^{7} T^{2} \)
13 \( 1 - 13108 T + p^{7} T^{2} \)
17 \( 1 - 40094 T + p^{7} T^{2} \)
19 \( 1 + p^{7} T^{2} \)
23 \( 1 + p^{7} T^{2} \)
29 \( 1 + 120844 T + p^{7} T^{2} \)
31 \( 1 + p^{7} T^{2} \)
37 \( 1 + 248316 T + p^{7} T^{2} \)
41 \( 1 + 9530 T + p^{7} T^{2} \)
43 \( 1 + p^{7} T^{2} \)
47 \( 1 + p^{7} T^{2} \)
53 \( 1 + 2015212 T + p^{7} T^{2} \)
59 \( 1 + p^{7} T^{2} \)
61 \( 1 + 532572 T + p^{7} T^{2} \)
67 \( 1 + p^{7} T^{2} \)
71 \( 1 + p^{7} T^{2} \)
73 \( 1 - 3917418 T + p^{7} T^{2} \)
79 \( 1 + p^{7} T^{2} \)
83 \( 1 + p^{7} T^{2} \)
89 \( 1 - 9246170 T + p^{7} T^{2} \)
97 \( 1 - 17567406 T + p^{7} T^{2} \)
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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

−10.63500213764716300505465416792, −9.052912878512082934936863716820, −8.146433261581757871265124392078, −7.68010767265190532009223347086, −6.30249339916178084884249549062, −5.09693169342656831294553461797, −3.64202246609979349419886742753, −3.30513563310468867474403829355, −1.11337320763100227832990700657, 0, 1.11337320763100227832990700657, 3.30513563310468867474403829355, 3.64202246609979349419886742753, 5.09693169342656831294553461797, 6.30249339916178084884249549062, 7.68010767265190532009223347086, 8.146433261581757871265124392078, 9.052912878512082934936863716820, 10.63500213764716300505465416792

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