Properties

Label 2-2e8-1.1-c1-0-4
Degree $2$
Conductor $256$
Sign $-1$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 6·11-s − 6·17-s − 2·19-s − 5·25-s + 4·27-s + 12·33-s + 6·41-s + 10·43-s − 7·49-s + 12·51-s + 4·57-s − 6·59-s + 14·67-s − 2·73-s + 10·75-s − 11·81-s − 18·83-s − 18·89-s + 10·97-s − 6·99-s − 6·107-s + 18·113-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.80·11-s − 1.45·17-s − 0.458·19-s − 25-s + 0.769·27-s + 2.08·33-s + 0.937·41-s + 1.52·43-s − 49-s + 1.68·51-s + 0.529·57-s − 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.15·75-s − 1.22·81-s − 1.97·83-s − 1.90·89-s + 1.01·97-s − 0.603·99-s − 0.580·107-s + 1.69·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-1$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{256} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−11.26097531340688560390760400117, −10.88783381933864566755233769368, −9.891997921642981576990976420061, −8.559961689771162774439116698021, −7.49638267192940041493128754308, −6.29051795613585298614624812000, −5.43412469058090981748265870176, −4.43148007247498493089277458199, −2.49619173311123854201331757855, 0, 2.49619173311123854201331757855, 4.43148007247498493089277458199, 5.43412469058090981748265870176, 6.29051795613585298614624812000, 7.49638267192940041493128754308, 8.559961689771162774439116698021, 9.891997921642981576990976420061, 10.88783381933864566755233769368, 11.26097531340688560390760400117

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