Properties

Degree $2$
Conductor $6012$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4.92·7-s + 2.15·11-s − 3.32·13-s − 5.91·17-s − 0.927·19-s + 7.63·23-s − 25-s + 1.59·29-s − 5.18·31-s − 9.84·35-s + 1.53·37-s + 9.71·41-s + 6.65·43-s − 5.72·47-s + 17.2·49-s + 9.24·53-s − 4.31·55-s − 5.78·59-s + 14.8·61-s + 6.65·65-s + 10.5·67-s − 5.51·71-s − 13.5·73-s + 10.6·77-s − 8.23·79-s − 7.97·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.85·7-s + 0.649·11-s − 0.922·13-s − 1.43·17-s − 0.212·19-s + 1.59·23-s − 0.200·25-s + 0.296·29-s − 0.930·31-s − 1.66·35-s + 0.251·37-s + 1.51·41-s + 1.01·43-s − 0.834·47-s + 2.45·49-s + 1.26·53-s − 0.581·55-s − 0.752·59-s + 1.90·61-s + 0.825·65-s + 1.29·67-s − 0.654·71-s − 1.58·73-s + 1.20·77-s − 0.926·79-s − 0.875·83-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: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981237037\)
\(L(\frac12)\) \(\approx\) \(1.981237037\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 5T^{2} \)
7 \( 1 - 4.92T + 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 + 0.927T + 19T^{2} \)
23 \( 1 - 7.63T + 23T^{2} \)
29 \( 1 - 1.59T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 - 1.53T + 37T^{2} \)
41 \( 1 - 9.71T + 41T^{2} \)
43 \( 1 - 6.65T + 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 + 5.78T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 8.23T + 79T^{2} \)
83 \( 1 + 7.97T + 83T^{2} \)
89 \( 1 - 8.41T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−8.043610436663286645378192510493, −7.32150259122230007069341973569, −7.02948998902466466320522221263, −5.84406734492128211482543189687, −4.97215536526650811547922277759, −4.45527393144613737258945312346, −3.94028393600128410470057689228, −2.64466689812934548514119343424, −1.86246907101348555792878078597, −0.75457279993641746987487924232, 0.75457279993641746987487924232, 1.86246907101348555792878078597, 2.64466689812934548514119343424, 3.94028393600128410470057689228, 4.45527393144613737258945312346, 4.97215536526650811547922277759, 5.84406734492128211482543189687, 7.02948998902466466320522221263, 7.32150259122230007069341973569, 8.043610436663286645378192510493

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