Properties

Label 2-6032-1.1-c1-0-57
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  − 3.16·3-s − 1.06·5-s − 3.52·7-s + 7.03·9-s − 2.93·11-s − 13-s + 3.38·15-s + 2.35·17-s − 2.71·19-s + 11.1·21-s + 1.38·23-s − 3.85·25-s − 12.7·27-s − 29-s + 3.37·31-s + 9.29·33-s + 3.77·35-s − 4.43·37-s + 3.16·39-s + 2.29·41-s + 1.23·43-s − 7.52·45-s + 4.44·47-s + 5.43·49-s − 7.47·51-s − 1.68·53-s + 3.13·55-s + ⋯
L(s)  = 1  − 1.82·3-s − 0.478·5-s − 1.33·7-s + 2.34·9-s − 0.884·11-s − 0.277·13-s + 0.874·15-s + 0.572·17-s − 0.621·19-s + 2.43·21-s + 0.288·23-s − 0.771·25-s − 2.46·27-s − 0.185·29-s + 0.605·31-s + 1.61·33-s + 0.637·35-s − 0.729·37-s + 0.507·39-s + 0.357·41-s + 0.187·43-s − 1.12·45-s + 0.647·47-s + 0.777·49-s − 1.04·51-s − 0.231·53-s + 0.422·55-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 \( 1 + 3.16T + 3T^{2} \)
5 \( 1 + 1.06T + 5T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 + 2.93T + 11T^{2} \)
17 \( 1 - 2.35T + 17T^{2} \)
19 \( 1 + 2.71T + 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + 4.43T + 37T^{2} \)
41 \( 1 - 2.29T + 41T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 + 1.68T + 53T^{2} \)
59 \( 1 + 3.67T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 4.07T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 8.61T + 79T^{2} \)
83 \( 1 - 2.27T + 83T^{2} \)
89 \( 1 - 7.46T + 89T^{2} \)
97 \( 1 + 8.96T + 97T^{2} \)
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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

−7.51103468939934683520842397240, −6.78082258178002433543432710253, −6.32794914237416366253156926256, −5.56932917665603247773172724287, −5.06762999023887820823067478839, −4.15566844047701363752574849611, −3.43340658929270120414534046372, −2.26093130050488857402466515823, −0.78232403494764473402700346996, 0, 0.78232403494764473402700346996, 2.26093130050488857402466515823, 3.43340658929270120414534046372, 4.15566844047701363752574849611, 5.06762999023887820823067478839, 5.56932917665603247773172724287, 6.32794914237416366253156926256, 6.78082258178002433543432710253, 7.51103468939934683520842397240

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