Properties

Label 2-8032-1.1-c1-0-220
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.12·3-s + 0.837·5-s + 0.914·7-s − 1.73·9-s − 2.68·11-s + 6.48·13-s + 0.941·15-s − 2.64·17-s − 2.38·19-s + 1.02·21-s + 1.02·23-s − 4.29·25-s − 5.32·27-s − 3.24·29-s − 0.564·31-s − 3.01·33-s + 0.766·35-s − 0.959·37-s + 7.28·39-s + 4.79·41-s − 6.94·43-s − 1.45·45-s − 6.03·47-s − 6.16·49-s − 2.96·51-s − 3.79·53-s − 2.25·55-s + ⋯
L(s)  = 1  + 0.648·3-s + 0.374·5-s + 0.345·7-s − 0.579·9-s − 0.810·11-s + 1.79·13-s + 0.243·15-s − 0.641·17-s − 0.546·19-s + 0.224·21-s + 0.213·23-s − 0.859·25-s − 1.02·27-s − 0.603·29-s − 0.101·31-s − 0.525·33-s + 0.129·35-s − 0.157·37-s + 1.16·39-s + 0.748·41-s − 1.05·43-s − 0.217·45-s − 0.880·47-s − 0.880·49-s − 0.415·51-s − 0.521·53-s − 0.303·55-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: \(1\)
Selberg data: \((2,\ 8032,\ (\ :1/2),\ -1)\)

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 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 1.12T + 3T^{2} \)
5 \( 1 - 0.837T + 5T^{2} \)
7 \( 1 - 0.914T + 7T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 2.38T + 19T^{2} \)
23 \( 1 - 1.02T + 23T^{2} \)
29 \( 1 + 3.24T + 29T^{2} \)
31 \( 1 + 0.564T + 31T^{2} \)
37 \( 1 + 0.959T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 + 6.03T + 47T^{2} \)
53 \( 1 + 3.79T + 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 - 3.45T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 0.544T + 71T^{2} \)
73 \( 1 - 5.94T + 73T^{2} \)
79 \( 1 - 3.81T + 79T^{2} \)
83 \( 1 + 4.21T + 83T^{2} \)
89 \( 1 - 3.26T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−7.79908850574935420609368507260, −6.71198971275129752755592943767, −6.05929161792859995847416224625, −5.51977320756569785442328015412, −4.62822257425201621478714645653, −3.74748590173055835406261933627, −3.11796839990553905484353298388, −2.20289086151091406299379947044, −1.51883688164584022791974427050, 0, 1.51883688164584022791974427050, 2.20289086151091406299379947044, 3.11796839990553905484353298388, 3.74748590173055835406261933627, 4.62822257425201621478714645653, 5.51977320756569785442328015412, 6.05929161792859995847416224625, 6.71198971275129752755592943767, 7.79908850574935420609368507260

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