Properties

Degree $2$
Conductor $8$
Sign $-1$
Motivic weight $19$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.26e4·3-s − 9.96e5·5-s − 7.24e7·7-s − 6.49e8·9-s + 4.59e9·11-s − 7.68e9·13-s − 2.25e10·15-s − 6.70e11·17-s − 6.03e11·19-s − 1.64e12·21-s − 1.41e13·23-s − 1.80e13·25-s − 4.10e13·27-s + 1.75e13·29-s + 7.97e13·31-s + 1.04e14·33-s + 7.22e13·35-s + 1.12e15·37-s − 1.73e14·39-s + 3.00e15·41-s − 3.41e15·43-s + 6.47e14·45-s − 1.14e16·47-s − 6.14e15·49-s − 1.51e16·51-s + 2.49e16·53-s − 4.58e15·55-s + ⋯
L(s)  = 1  + 0.664·3-s − 0.228·5-s − 0.678·7-s − 0.559·9-s + 0.587·11-s − 0.201·13-s − 0.151·15-s − 1.37·17-s − 0.429·19-s − 0.450·21-s − 1.63·23-s − 0.947·25-s − 1.03·27-s + 0.224·29-s + 0.541·31-s + 0.390·33-s + 0.154·35-s + 1.41·37-s − 0.133·39-s + 1.43·41-s − 1.03·43-s + 0.127·45-s − 1.49·47-s − 0.539·49-s − 0.910·51-s + 1.03·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-1$
Motivic weight: \(19\)
Character: $\chi_{8} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8,\ (\ :19/2),\ -1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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 \( 1 - 2.26e4T + 1.16e9T^{2} \)
5 \( 1 + 9.96e5T + 1.90e13T^{2} \)
7 \( 1 + 7.24e7T + 1.13e16T^{2} \)
11 \( 1 - 4.59e9T + 6.11e19T^{2} \)
13 \( 1 + 7.68e9T + 1.46e21T^{2} \)
17 \( 1 + 6.70e11T + 2.39e23T^{2} \)
19 \( 1 + 6.03e11T + 1.97e24T^{2} \)
23 \( 1 + 1.41e13T + 7.46e25T^{2} \)
29 \( 1 - 1.75e13T + 6.10e27T^{2} \)
31 \( 1 - 7.97e13T + 2.16e28T^{2} \)
37 \( 1 - 1.12e15T + 6.24e29T^{2} \)
41 \( 1 - 3.00e15T + 4.39e30T^{2} \)
43 \( 1 + 3.41e15T + 1.08e31T^{2} \)
47 \( 1 + 1.14e16T + 5.88e31T^{2} \)
53 \( 1 - 2.49e16T + 5.77e32T^{2} \)
59 \( 1 - 7.22e15T + 4.42e33T^{2} \)
61 \( 1 - 1.29e17T + 8.34e33T^{2} \)
67 \( 1 - 2.74e17T + 4.95e34T^{2} \)
71 \( 1 - 1.56e17T + 1.49e35T^{2} \)
73 \( 1 + 8.21e17T + 2.53e35T^{2} \)
79 \( 1 - 2.09e17T + 1.13e36T^{2} \)
83 \( 1 - 3.15e17T + 2.90e36T^{2} \)
89 \( 1 - 2.78e18T + 1.09e37T^{2} \)
97 \( 1 - 7.58e18T + 5.60e37T^{2} \)
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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

−16.06603715345740463686071880437, −14.64888542996517703733943344671, −13.30733167267753409188456694186, −11.59222335920846488912957148529, −9.624696113091132773015213100438, −8.221718538408028992811871784240, −6.32387363019394249336450281071, −3.97188059486045678904331918274, −2.35086668857613831103081000435, 0, 2.35086668857613831103081000435, 3.97188059486045678904331918274, 6.32387363019394249336450281071, 8.221718538408028992811871784240, 9.624696113091132773015213100438, 11.59222335920846488912957148529, 13.30733167267753409188456694186, 14.64888542996517703733943344671, 16.06603715345740463686071880437

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