Properties

Label 2-6032-1.1-c1-0-66
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.06·3-s + 1.60·5-s − 0.812·7-s − 1.87·9-s + 5.39·11-s − 13-s + 1.69·15-s + 3.51·17-s − 3.73·19-s − 0.861·21-s − 7.75·23-s − 2.43·25-s − 5.16·27-s + 29-s + 6.56·31-s + 5.71·33-s − 1.30·35-s − 3.76·37-s − 1.06·39-s + 8.55·41-s + 12.8·43-s − 3.00·45-s + 11.4·47-s − 6.33·49-s + 3.72·51-s + 6.10·53-s + 8.63·55-s + ⋯
L(s)  = 1  + 0.612·3-s + 0.716·5-s − 0.307·7-s − 0.625·9-s + 1.62·11-s − 0.277·13-s + 0.438·15-s + 0.853·17-s − 0.856·19-s − 0.188·21-s − 1.61·23-s − 0.487·25-s − 0.994·27-s + 0.185·29-s + 1.17·31-s + 0.995·33-s − 0.219·35-s − 0.619·37-s − 0.169·39-s + 1.33·41-s + 1.96·43-s − 0.447·45-s + 1.66·47-s − 0.905·49-s + 0.522·51-s + 0.838·53-s + 1.16·55-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: \(0\)
Selberg data: \((2,\ 6032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.837679171\)
\(L(\frac12)\) \(\approx\) \(2.837679171\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.06T + 3T^{2} \)
5 \( 1 - 1.60T + 5T^{2} \)
7 \( 1 + 0.812T + 7T^{2} \)
11 \( 1 - 5.39T + 11T^{2} \)
17 \( 1 - 3.51T + 17T^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + 7.75T + 23T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 3.76T + 37T^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 - 7.47T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 - 5.26T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 2.24T + 73T^{2} \)
79 \( 1 + 9.67T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 5.02T + 97T^{2} \)
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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

−8.162505339282345710490072116477, −7.46458762271497884216937725217, −6.45514959062713846677335368801, −6.07390525612503519737959983964, −5.40503547007670400422489081147, −4.06512587813723036745326761039, −3.82311755206424671056035840296, −2.59179382035690188544224684882, −2.07669820378201200055141088732, −0.864626626947482912709667097409, 0.864626626947482912709667097409, 2.07669820378201200055141088732, 2.59179382035690188544224684882, 3.82311755206424671056035840296, 4.06512587813723036745326761039, 5.40503547007670400422489081147, 6.07390525612503519737959983964, 6.45514959062713846677335368801, 7.46458762271497884216937725217, 8.162505339282345710490072116477

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