Properties

Label 2-8092-1.1-c1-0-133
Degree $2$
Conductor $8092$
Sign $-1$
Analytic cond. $64.6149$
Root an. cond. $8.03834$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·3-s + 1.30·5-s + 7-s + 2.30·9-s − 4·11-s − 4.60·13-s + 3·15-s + 8.60·19-s + 2.30·21-s − 4·23-s − 3.30·25-s − 1.60·27-s − 9.21·29-s − 7.30·31-s − 9.21·33-s + 1.30·35-s − 9.81·37-s − 10.6·39-s + 11.5·41-s − 4.30·43-s + 3.00·45-s − 2.60·47-s + 49-s + 0.697·53-s − 5.21·55-s + 19.8·57-s − 8·59-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.582·5-s + 0.377·7-s + 0.767·9-s − 1.20·11-s − 1.27·13-s + 0.774·15-s + 1.97·19-s + 0.502·21-s − 0.834·23-s − 0.660·25-s − 0.308·27-s − 1.71·29-s − 1.31·31-s − 1.60·33-s + 0.220·35-s − 1.61·37-s − 1.69·39-s + 1.79·41-s − 0.656·43-s + 0.447·45-s − 0.380·47-s + 0.142·49-s + 0.0957·53-s − 0.702·55-s + 2.62·57-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8092\)    =    \(2^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(64.6149\)
Root analytic conductor: \(8.03834\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8092,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good3 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
19 \( 1 - 8.60T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 + 9.81T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 - 0.697T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 - 2.69T + 67T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 + 7.51T + 73T^{2} \)
79 \( 1 - 2.60T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−7.50957415091721636415570775408, −7.34780139711844026350938227739, −5.85551532866937145647957735204, −5.42265490053098422843818840970, −4.71231873130702618614145652308, −3.60714235422820093751129655167, −3.06367076501529526924961044075, −2.17232126102681910679320784911, −1.76705197247416741712539884751, 0, 1.76705197247416741712539884751, 2.17232126102681910679320784911, 3.06367076501529526924961044075, 3.60714235422820093751129655167, 4.71231873130702618614145652308, 5.42265490053098422843818840970, 5.85551532866937145647957735204, 7.34780139711844026350938227739, 7.50957415091721636415570775408

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