Properties

Degree $2$
Conductor $8$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s + 24·7-s − 11·9-s − 44·11-s + 22·13-s + 8·15-s + 50·17-s + 44·19-s − 96·21-s − 56·23-s − 121·25-s + 152·27-s + 198·29-s − 160·31-s + 176·33-s − 48·35-s − 162·37-s − 88·39-s − 198·41-s + 52·43-s + 22·45-s + 528·47-s + 233·49-s − 200·51-s − 242·53-s + 88·55-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.178·5-s + 1.29·7-s − 0.407·9-s − 1.20·11-s + 0.469·13-s + 0.137·15-s + 0.713·17-s + 0.531·19-s − 0.997·21-s − 0.507·23-s − 0.967·25-s + 1.08·27-s + 1.26·29-s − 0.926·31-s + 0.928·33-s − 0.231·35-s − 0.719·37-s − 0.361·39-s − 0.754·41-s + 0.184·43-s + 0.0728·45-s + 1.63·47-s + 0.679·49-s − 0.549·51-s − 0.627·53-s + 0.215·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{8} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.690031\)
\(L(\frac12)\) \(\approx\) \(0.690031\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
5 \( 1 + 2 T + p^{3} T^{2} \)
7 \( 1 - 24 T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 - 50 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 + 56 T + p^{3} T^{2} \)
29 \( 1 - 198 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 + 198 T + p^{3} T^{2} \)
43 \( 1 - 52 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 + 242 T + p^{3} T^{2} \)
59 \( 1 + 668 T + p^{3} T^{2} \)
61 \( 1 - 550 T + p^{3} T^{2} \)
67 \( 1 - 188 T + p^{3} T^{2} \)
71 \( 1 - 728 T + p^{3} T^{2} \)
73 \( 1 - 154 T + p^{3} T^{2} \)
79 \( 1 + 656 T + p^{3} T^{2} \)
83 \( 1 - 236 T + p^{3} T^{2} \)
89 \( 1 - 714 T + p^{3} T^{2} \)
97 \( 1 + 478 T + p^{3} T^{2} \)
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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

−21.58285764863616869137931013887, −20.39967859251983642348124594597, −18.40188539570615183142109814395, −17.39459794290390525945222243855, −15.81222026485925572675714003132, −14.06576395821067509542343046445, −11.97018150237372979238940257079, −10.71326597252919103933711342457, −8.041114144439934942917179504468, −5.36694472262759880467468382418, 5.36694472262759880467468382418, 8.041114144439934942917179504468, 10.71326597252919103933711342457, 11.97018150237372979238940257079, 14.06576395821067509542343046445, 15.81222026485925572675714003132, 17.39459794290390525945222243855, 18.40188539570615183142109814395, 20.39967859251983642348124594597, 21.58285764863616869137931013887

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