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  − 0.539·5-s + 1.07·7-s − 3.70·11-s + 4.34·13-s + 1.46·17-s + 3.41·19-s − 8.68·23-s − 4.70·25-s − 2.15·29-s − 1.26·31-s − 0.581·35-s + 11.7·37-s − 9.95·41-s − 9.80·43-s + 7.70·47-s − 5.83·49-s + 9.55·53-s + 2·55-s − 6.92·59-s + 1.70·61-s − 2.34·65-s − 0.879·67-s − 4·71-s + 1.41·73-s − 4·77-s − 15.8·79-s + 0.183·83-s + ⋯
L(s)  = 1  − 0.241·5-s + 0.407·7-s − 1.11·11-s + 1.20·13-s + 0.354·17-s + 0.784·19-s − 1.80·23-s − 0.941·25-s − 0.400·29-s − 0.226·31-s − 0.0982·35-s + 1.93·37-s − 1.55·41-s − 1.49·43-s + 1.12·47-s − 0.833·49-s + 1.31·53-s + 0.269·55-s − 0.901·59-s + 0.218·61-s − 0.290·65-s − 0.107·67-s − 0.474·71-s + 0.166·73-s − 0.455·77-s − 1.78·79-s + 0.0201·83-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 \( 1 + 0.539T + 5T^{2} \)
7 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 - 1.46T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 + 8.68T + 23T^{2} \)
29 \( 1 + 2.15T + 29T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + 9.95T + 41T^{2} \)
43 \( 1 + 9.80T + 43T^{2} \)
47 \( 1 - 7.70T + 47T^{2} \)
53 \( 1 - 9.55T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 1.70T + 61T^{2} \)
67 \( 1 + 0.879T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 0.183T + 83T^{2} \)
89 \( 1 + 6.15T + 89T^{2} \)
97 \( 1 - 9.23T + 97T^{2} \)
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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

−7.912193074659716619030963446180, −7.19069949958222107642056482055, −6.07535659185797601654896217454, −5.71545693289591363208641375503, −4.84708064954987266911537640589, −3.97544583953630139217965415944, −3.33035234859659137801959153193, −2.28197009010112592803274883800, −1.36162988100844992625867441643, 0, 1.36162988100844992625867441643, 2.28197009010112592803274883800, 3.33035234859659137801959153193, 3.97544583953630139217965415944, 4.84708064954987266911537640589, 5.71545693289591363208641375503, 6.07535659185797601654896217454, 7.19069949958222107642056482055, 7.912193074659716619030963446180

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