Properties

Label 2-8032-1.1-c1-0-229
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  + 2.26·3-s − 0.762·5-s + 1.25·7-s + 2.12·9-s − 1.90·11-s − 2.67·13-s − 1.72·15-s + 7.34·17-s − 6.08·19-s + 2.84·21-s − 0.363·23-s − 4.41·25-s − 1.98·27-s + 8.19·29-s − 5.80·31-s − 4.31·33-s − 0.957·35-s − 6.69·37-s − 6.05·39-s − 5.03·41-s − 8.74·43-s − 1.61·45-s + 7.61·47-s − 5.42·49-s + 16.6·51-s − 3.07·53-s + 1.45·55-s + ⋯
L(s)  = 1  + 1.30·3-s − 0.341·5-s + 0.474·7-s + 0.707·9-s − 0.575·11-s − 0.741·13-s − 0.445·15-s + 1.78·17-s − 1.39·19-s + 0.620·21-s − 0.0758·23-s − 0.883·25-s − 0.381·27-s + 1.52·29-s − 1.04·31-s − 0.751·33-s − 0.161·35-s − 1.09·37-s − 0.969·39-s − 0.785·41-s − 1.33·43-s − 0.241·45-s + 1.11·47-s − 0.774·49-s + 2.32·51-s − 0.422·53-s + 0.196·55-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 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 + 0.762T + 5T^{2} \)
7 \( 1 - 1.25T + 7T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + 6.08T + 19T^{2} \)
23 \( 1 + 0.363T + 23T^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + 6.69T + 37T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 + 8.74T + 43T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 + 3.07T + 53T^{2} \)
59 \( 1 + 5.91T + 59T^{2} \)
61 \( 1 + 3.36T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 5.45T + 71T^{2} \)
73 \( 1 - 7.16T + 73T^{2} \)
79 \( 1 - 9.28T + 79T^{2} \)
83 \( 1 + 1.97T + 83T^{2} \)
89 \( 1 + 8.78T + 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−7.76026988908367603006894227927, −7.09233558116051129300185390527, −6.12782757471355692159670023434, −5.24050562371929432530448194996, −4.61139757652163126560836970145, −3.65898049841205355434532254240, −3.15803230560114818656378382599, −2.28616119683050728339940022747, −1.57831998617813243847809668765, 0, 1.57831998617813243847809668765, 2.28616119683050728339940022747, 3.15803230560114818656378382599, 3.65898049841205355434532254240, 4.61139757652163126560836970145, 5.24050562371929432530448194996, 6.12782757471355692159670023434, 7.09233558116051129300185390527, 7.76026988908367603006894227927

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