Properties

Label 2-6032-1.1-c1-0-145
Degree $2$
Conductor $6032$
Sign $-1$
Analytic cond. $48.1657$
Root an. cond. $6.94015$
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  + 2.55·3-s − 2.69·5-s + 1.01·7-s + 3.53·9-s − 2.93·11-s − 13-s − 6.88·15-s − 4.17·17-s + 1.18·19-s + 2.58·21-s + 6.85·23-s + 2.26·25-s + 1.36·27-s − 29-s + 4.62·31-s − 7.50·33-s − 2.72·35-s − 4.42·37-s − 2.55·39-s + 5.14·41-s − 3.08·43-s − 9.51·45-s + 9.02·47-s − 5.97·49-s − 10.6·51-s − 6.85·53-s + 7.91·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 1.20·5-s + 0.381·7-s + 1.17·9-s − 0.885·11-s − 0.277·13-s − 1.77·15-s − 1.01·17-s + 0.271·19-s + 0.563·21-s + 1.42·23-s + 0.452·25-s + 0.261·27-s − 0.185·29-s + 0.830·31-s − 1.30·33-s − 0.460·35-s − 0.727·37-s − 0.409·39-s + 0.804·41-s − 0.470·43-s − 1.41·45-s + 1.31·47-s − 0.854·49-s − 1.49·51-s − 0.941·53-s + 1.06·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6032\)    =    \(2^{4} \cdot 13 \cdot 29\)
Sign: $-1$
Analytic conductor: \(48.1657\)
Root analytic conductor: \(6.94015\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6032,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
29 \( 1 + T \)
good3 \( 1 - 2.55T + 3T^{2} \)
5 \( 1 + 2.69T + 5T^{2} \)
7 \( 1 - 1.01T + 7T^{2} \)
11 \( 1 + 2.93T + 11T^{2} \)
17 \( 1 + 4.17T + 17T^{2} \)
19 \( 1 - 1.18T + 19T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 - 9.02T + 47T^{2} \)
53 \( 1 + 6.85T + 53T^{2} \)
59 \( 1 + 4.90T + 59T^{2} \)
61 \( 1 + 7.80T + 61T^{2} \)
67 \( 1 - 1.60T + 67T^{2} \)
71 \( 1 + 1.51T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 7.59T + 79T^{2} \)
83 \( 1 + 1.23T + 83T^{2} \)
89 \( 1 - 8.99T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.76815300939254185523574151920, −7.40075612414348714120986984520, −6.61873088444712429502452329258, −5.35319973164817794161698120725, −4.55797604006326488795227079679, −4.00920096882246790783201759284, −3.01441648757327375532802742651, −2.67984053317663685094997810118, −1.51016222367431392740855512969, 0, 1.51016222367431392740855512969, 2.67984053317663685094997810118, 3.01441648757327375532802742651, 4.00920096882246790783201759284, 4.55797604006326488795227079679, 5.35319973164817794161698120725, 6.61873088444712429502452329258, 7.40075612414348714120986984520, 7.76815300939254185523574151920

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