Properties

Label 2-2e8-1.1-c1-0-5
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  − 4·5-s − 3·9-s − 4·13-s − 2·17-s + 11·25-s − 4·29-s + 12·37-s − 10·41-s + 12·45-s − 7·49-s − 4·53-s + 12·61-s + 16·65-s − 6·73-s + 9·81-s + 8·85-s + 10·89-s − 18·97-s − 20·101-s − 20·109-s − 14·113-s + 12·117-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s − 1.10·13-s − 0.485·17-s + 11/5·25-s − 0.742·29-s + 1.97·37-s − 1.56·41-s + 1.78·45-s − 49-s − 0.549·53-s + 1.53·61-s + 1.98·65-s − 0.702·73-s + 81-s + 0.867·85-s + 1.05·89-s − 1.82·97-s − 1.99·101-s − 1.91·109-s − 1.31·113-s + 1.10·117-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 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−11.56957687474889255866293555846, −10.93612266887391281534482819402, −9.531816445825083349512041454382, −8.389848986528649416792303765934, −7.74661145077041678977550300968, −6.71650754317349099241460586550, −5.15538859196444031818760258484, −4.06135513439308426356140505035, −2.84412696583825002808260003434, 0, 2.84412696583825002808260003434, 4.06135513439308426356140505035, 5.15538859196444031818760258484, 6.71650754317349099241460586550, 7.74661145077041678977550300968, 8.389848986528649416792303765934, 9.531816445825083349512041454382, 10.93612266887391281534482819402, 11.56957687474889255866293555846

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