Properties

Label 2-8032-1.1-c1-0-103
Degree $2$
Conductor $8032$
Sign $1$
Analytic cond. $64.1358$
Root an. cond. $8.00848$
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  + 0.369·3-s − 3.94·5-s + 3.05·7-s − 2.86·9-s + 5.92·11-s + 4.39·13-s − 1.45·15-s + 6.07·17-s + 1.60·19-s + 1.12·21-s + 7.25·23-s + 10.5·25-s − 2.16·27-s + 2.83·29-s − 6.95·31-s + 2.18·33-s − 12.0·35-s + 7.42·37-s + 1.62·39-s − 3.32·41-s + 1.53·43-s + 11.2·45-s − 3.21·47-s + 2.34·49-s + 2.24·51-s − 9.95·53-s − 23.3·55-s + ⋯
L(s)  = 1  + 0.213·3-s − 1.76·5-s + 1.15·7-s − 0.954·9-s + 1.78·11-s + 1.21·13-s − 0.376·15-s + 1.47·17-s + 0.368·19-s + 0.246·21-s + 1.51·23-s + 2.11·25-s − 0.416·27-s + 0.525·29-s − 1.24·31-s + 0.381·33-s − 2.03·35-s + 1.22·37-s + 0.260·39-s − 0.519·41-s + 0.233·43-s + 1.68·45-s − 0.469·47-s + 0.335·49-s + 0.314·51-s − 1.36·53-s − 3.15·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8032\)    =    \(2^{5} \cdot 251\)
Sign: $1$
Analytic conductor: \(64.1358\)
Root analytic conductor: \(8.00848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.338055552\)
\(L(\frac12)\) \(\approx\) \(2.338055552\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 0.369T + 3T^{2} \)
5 \( 1 + 3.94T + 5T^{2} \)
7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 - 5.92T + 11T^{2} \)
13 \( 1 - 4.39T + 13T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 - 7.25T + 23T^{2} \)
29 \( 1 - 2.83T + 29T^{2} \)
31 \( 1 + 6.95T + 31T^{2} \)
37 \( 1 - 7.42T + 37T^{2} \)
41 \( 1 + 3.32T + 41T^{2} \)
43 \( 1 - 1.53T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 + 9.95T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 - 9.96T + 61T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 - 9.26T + 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 - 18.0T + 83T^{2} \)
89 \( 1 + 8.06T + 89T^{2} \)
97 \( 1 - 0.446T + 97T^{2} \)
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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

−8.103327806969307558723052128091, −7.26223588067615973080533906470, −6.61573663653969096813845028826, −5.67536744241823674744483171938, −4.93060993429674939085976949145, −4.10825252450404377280849691842, −3.55154072647261150721123561439, −3.04472898246792798159785809344, −1.44060323797923623622684643978, −0.864117273812423011919062680864, 0.864117273812423011919062680864, 1.44060323797923623622684643978, 3.04472898246792798159785809344, 3.55154072647261150721123561439, 4.10825252450404377280849691842, 4.93060993429674939085976949145, 5.67536744241823674744483171938, 6.61573663653969096813845028826, 7.26223588067615973080533906470, 8.103327806969307558723052128091

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