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  − 2.21·5-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: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2142826121\)
\(L(\frac12)\) \(\approx\) \(0.2142826121\)
\(L(\frac{3}{2})\) 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

−8.175797488033433909104912530868, −7.14776912905252219079173761069, −6.82169452793044580027582647314, −6.07209263457006207690367153371, −5.25107627088333417742413073180, −4.11201258441599326333491277896, −3.77354136993346645666045609266, −2.92676938673234464202461887227, −1.96938113176195731561375231527, −0.22611135603423046277344123224, 0.22611135603423046277344123224, 1.96938113176195731561375231527, 2.92676938673234464202461887227, 3.77354136993346645666045609266, 4.11201258441599326333491277896, 5.25107627088333417742413073180, 6.07209263457006207690367153371, 6.82169452793044580027582647314, 7.14776912905252219079173761069, 8.175797488033433909104912530868

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