Properties

Label 2-6032-1.1-c1-0-3
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  − 2.22·3-s − 0.143·5-s − 1.44·7-s + 1.92·9-s − 4.98·11-s + 13-s + 0.317·15-s − 2.87·17-s − 1.75·19-s + 3.20·21-s − 2.19·23-s − 4.97·25-s + 2.37·27-s − 29-s − 10.4·31-s + 11.0·33-s + 0.206·35-s − 3.76·37-s − 2.22·39-s + 9.68·41-s − 8.39·43-s − 0.276·45-s − 0.154·47-s − 4.91·49-s + 6.38·51-s + 1.92·53-s + 0.713·55-s + ⋯
L(s)  = 1  − 1.28·3-s − 0.0640·5-s − 0.545·7-s + 0.643·9-s − 1.50·11-s + 0.277·13-s + 0.0820·15-s − 0.697·17-s − 0.403·19-s + 0.699·21-s − 0.456·23-s − 0.995·25-s + 0.457·27-s − 0.185·29-s − 1.87·31-s + 1.92·33-s + 0.0349·35-s − 0.619·37-s − 0.355·39-s + 1.51·41-s − 1.28·43-s − 0.0411·45-s − 0.0225·47-s − 0.702·49-s + 0.894·51-s + 0.264·53-s + 0.0961·55-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

\(L(1)\) \(\approx\) \(0.1591545081\)
\(L(\frac12)\) \(\approx\) \(0.1591545081\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.22T + 3T^{2} \)
5 \( 1 + 0.143T + 5T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + 4.98T + 11T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 + 1.75T + 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 3.76T + 37T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 + 8.39T + 43T^{2} \)
47 \( 1 + 0.154T + 47T^{2} \)
53 \( 1 - 1.92T + 53T^{2} \)
59 \( 1 - 4.22T + 59T^{2} \)
61 \( 1 + 0.298T + 61T^{2} \)
67 \( 1 + 0.522T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 5.40T + 73T^{2} \)
79 \( 1 + 1.00T + 79T^{2} \)
83 \( 1 + 0.385T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 5.94T + 97T^{2} \)
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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

−7.981584225173814579799831499228, −7.23226952044233179656225994783, −6.56533415432838785753815836900, −5.75687984211562368882494766216, −5.47417609368667665211483509427, −4.59272026029912408265017396170, −3.76875960619587348244139646366, −2.75017289902612941420918893970, −1.77431048359625991231096707122, −0.21276734828278822179572073195, 0.21276734828278822179572073195, 1.77431048359625991231096707122, 2.75017289902612941420918893970, 3.76875960619587348244139646366, 4.59272026029912408265017396170, 5.47417609368667665211483509427, 5.75687984211562368882494766216, 6.56533415432838785753815836900, 7.23226952044233179656225994783, 7.981584225173814579799831499228

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