Properties

Label 2-6032-1.1-c1-0-72
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  − 3.05·3-s − 3.01·5-s + 1.81·7-s + 6.34·9-s + 2.20·11-s + 13-s + 9.22·15-s + 0.141·17-s − 6.46·19-s − 5.54·21-s − 7.71·23-s + 4.10·25-s − 10.2·27-s + 29-s + 6.97·31-s − 6.73·33-s − 5.47·35-s − 2.01·37-s − 3.05·39-s + 12.4·41-s − 5.60·43-s − 19.1·45-s − 7.19·47-s − 3.70·49-s − 0.431·51-s + 2.34·53-s − 6.65·55-s + ⋯
L(s)  = 1  − 1.76·3-s − 1.34·5-s + 0.685·7-s + 2.11·9-s + 0.664·11-s + 0.277·13-s + 2.38·15-s + 0.0342·17-s − 1.48·19-s − 1.20·21-s − 1.60·23-s + 0.820·25-s − 1.96·27-s + 0.185·29-s + 1.25·31-s − 1.17·33-s − 0.925·35-s − 0.332·37-s − 0.489·39-s + 1.94·41-s − 0.854·43-s − 2.85·45-s − 1.04·47-s − 0.529·49-s − 0.0604·51-s + 0.322·53-s − 0.896·55-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 \( 1 + 3.05T + 3T^{2} \)
5 \( 1 + 3.01T + 5T^{2} \)
7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
17 \( 1 - 0.141T + 17T^{2} \)
19 \( 1 + 6.46T + 19T^{2} \)
23 \( 1 + 7.71T + 23T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 + 2.01T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 - 2.34T + 53T^{2} \)
59 \( 1 - 4.14T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 + 2.56T + 73T^{2} \)
79 \( 1 - 7.81T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 9.59T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.84516193289627605301245178992, −6.71369649150029132776092440923, −6.41620345823864813233893667698, −5.62549305361648513211103109519, −4.64972078500792029738434616470, −4.32417447501774628938992572029, −3.65874790733382563888609722848, −2.03730183275126946361421262214, −0.945296696636214778323877324570, 0, 0.945296696636214778323877324570, 2.03730183275126946361421262214, 3.65874790733382563888609722848, 4.32417447501774628938992572029, 4.64972078500792029738434616470, 5.62549305361648513211103109519, 6.41620345823864813233893667698, 6.71369649150029132776092440923, 7.84516193289627605301245178992

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