Properties

Label 2-8032-1.1-c1-0-4
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  − 1.62·3-s − 2.98·5-s − 1.58·7-s − 0.374·9-s + 1.17·11-s − 1.64·13-s + 4.84·15-s − 0.163·17-s − 0.993·19-s + 2.56·21-s − 3.53·23-s + 3.93·25-s + 5.46·27-s − 0.843·29-s − 7.13·31-s − 1.90·33-s + 4.73·35-s + 5.41·37-s + 2.66·39-s − 0.419·41-s − 3.12·43-s + 1.11·45-s − 12.0·47-s − 4.49·49-s + 0.264·51-s − 7.57·53-s − 3.51·55-s + ⋯
L(s)  = 1  − 0.935·3-s − 1.33·5-s − 0.598·7-s − 0.124·9-s + 0.354·11-s − 0.455·13-s + 1.25·15-s − 0.0395·17-s − 0.227·19-s + 0.559·21-s − 0.737·23-s + 0.786·25-s + 1.05·27-s − 0.156·29-s − 1.28·31-s − 0.331·33-s + 0.800·35-s + 0.889·37-s + 0.426·39-s − 0.0655·41-s − 0.476·43-s + 0.166·45-s − 1.75·47-s − 0.641·49-s + 0.0370·51-s − 1.04·53-s − 0.474·55-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

\(L(1)\) \(\approx\) \(0.05863491482\)
\(L(\frac12)\) \(\approx\) \(0.05863491482\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.62T + 3T^{2} \)
5 \( 1 + 2.98T + 5T^{2} \)
7 \( 1 + 1.58T + 7T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
17 \( 1 + 0.163T + 17T^{2} \)
19 \( 1 + 0.993T + 19T^{2} \)
23 \( 1 + 3.53T + 23T^{2} \)
29 \( 1 + 0.843T + 29T^{2} \)
31 \( 1 + 7.13T + 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 0.419T + 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 + 6.22T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + 1.54T + 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + 9.05T + 73T^{2} \)
79 \( 1 - 5.44T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.81118530839379640330005069497, −7.09067079946422646124643185301, −6.39825917153610352543090194259, −5.88277882875317736571466810387, −4.94302192403671309335639050917, −4.37723328293733095613872103982, −3.55291281882381447585408170533, −2.91317641498704423188178567052, −1.56437242989173278639285277309, −0.12503113982004682494752560533, 0.12503113982004682494752560533, 1.56437242989173278639285277309, 2.91317641498704423188178567052, 3.55291281882381447585408170533, 4.37723328293733095613872103982, 4.94302192403671309335639050917, 5.88277882875317736571466810387, 6.39825917153610352543090194259, 7.09067079946422646124643185301, 7.81118530839379640330005069497

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