Properties

Degree $2$
Conductor $6012$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.21·5-s − 4.42·7-s + 0.903·11-s + 0.622·13-s + 4.21·17-s − 5.80·19-s − 1.24·23-s − 0.0967·25-s + 8.85·29-s − 3.05·31-s − 9.80·35-s − 1.18·37-s + 2.02·41-s − 8.83·43-s + 3.09·47-s + 12.6·49-s − 4.34·53-s + 2·55-s − 12.4·59-s − 2.90·61-s + 1.37·65-s + 5.59·67-s − 4·71-s − 7.80·73-s − 4·77-s + 1.72·79-s + 7.47·83-s + ⋯
L(s)  = 1  + 0.990·5-s − 1.67·7-s + 0.272·11-s + 0.172·13-s + 1.02·17-s − 1.33·19-s − 0.259·23-s − 0.0193·25-s + 1.64·29-s − 0.547·31-s − 1.65·35-s − 0.194·37-s + 0.315·41-s − 1.34·43-s + 0.451·47-s + 1.80·49-s − 0.597·53-s + 0.269·55-s − 1.61·59-s − 0.371·61-s + 0.170·65-s + 0.683·67-s − 0.474·71-s − 0.913·73-s − 0.455·77-s + 0.194·79-s + 0.820·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6012} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 2.21T + 5T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 - 0.903T + 11T^{2} \)
13 \( 1 - 0.622T + 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 - 8.85T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + 1.18T + 37T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 8.83T + 43T^{2} \)
47 \( 1 - 3.09T + 47T^{2} \)
53 \( 1 + 4.34T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 2.90T + 61T^{2} \)
67 \( 1 - 5.59T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 7.80T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 - 7.47T + 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 + 7.28T + 97T^{2} \)
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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

−7.69351671241809048579920522787, −6.71674303040775610606304936155, −6.29930270981859626091961289981, −5.86112989580193061797586144262, −4.90763486570166228243254261752, −3.90221585893798248416976063916, −3.17654083489179157909077119095, −2.41515282276789002075468673689, −1.36158697416898819048187976333, 0, 1.36158697416898819048187976333, 2.41515282276789002075468673689, 3.17654083489179157909077119095, 3.90221585893798248416976063916, 4.90763486570166228243254261752, 5.86112989580193061797586144262, 6.29930270981859626091961289981, 6.71674303040775610606304936155, 7.69351671241809048579920522787

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