Properties

Label 2-2e9-1.1-c1-0-1
Degree $2$
Conductor $512$
Sign $1$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 2.82·5-s + 4·7-s − 0.999·9-s − 1.41·11-s + 2.82·13-s + 4.00·15-s − 4·17-s + 7.07·19-s − 5.65·21-s + 4·23-s + 3.00·25-s + 5.65·27-s + 8.48·29-s + 8·31-s + 2.00·33-s − 11.3·35-s + 2.82·37-s − 4.00·39-s + 2·41-s − 4.24·43-s + 2.82·45-s + 9·49-s + 5.65·51-s + 2.82·53-s + 4.00·55-s − 10.0·57-s + ⋯
L(s)  = 1  − 0.816·3-s − 1.26·5-s + 1.51·7-s − 0.333·9-s − 0.426·11-s + 0.784·13-s + 1.03·15-s − 0.970·17-s + 1.62·19-s − 1.23·21-s + 0.834·23-s + 0.600·25-s + 1.08·27-s + 1.57·29-s + 1.43·31-s + 0.348·33-s − 1.91·35-s + 0.464·37-s − 0.640·39-s + 0.312·41-s − 0.646·43-s + 0.421·45-s + 1.28·49-s + 0.792·51-s + 0.388·53-s + 0.539·55-s − 1.32·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $1$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9755964828\)
\(L(\frac12)\) \(\approx\) \(0.9755964828\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 + 4.24T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−11.16864900599586194902434800352, −10.42637178033083658237172163282, −8.813439232030150242587399693514, −8.167058260342329260108230696044, −7.41511515077245583457173870416, −6.23489109353507760925346992833, −5.02210215319635069469174047645, −4.49960645185009216472236312771, −3.00436965059298781127362087828, −0.976243472162187210614835272268, 0.976243472162187210614835272268, 3.00436965059298781127362087828, 4.49960645185009216472236312771, 5.02210215319635069469174047645, 6.23489109353507760925346992833, 7.41511515077245583457173870416, 8.167058260342329260108230696044, 8.813439232030150242587399693514, 10.42637178033083658237172163282, 11.16864900599586194902434800352

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