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  + 1.67·5-s + 3.35·7-s + 1.19·11-s − 2.96·13-s − 0.324·17-s − 1.61·19-s − 5.92·23-s − 2.19·25-s + 6.70·29-s + 8.31·31-s + 5.61·35-s − 0.574·37-s + 6.06·41-s − 1.36·43-s − 5.19·47-s + 4.22·49-s + 11.2·53-s + 2·55-s + 4.64·59-s − 0.806·61-s − 4.96·65-s + 5.28·67-s + 4·71-s − 3.61·73-s + 4·77-s + 12.1·79-s + 11.6·83-s + ⋯
L(s)  = 1  + 0.749·5-s + 1.26·7-s + 0.359·11-s − 0.821·13-s − 0.0787·17-s − 0.369·19-s − 1.23·23-s − 0.438·25-s + 1.24·29-s + 1.49·31-s + 0.948·35-s − 0.0944·37-s + 0.946·41-s − 0.207·43-s − 0.757·47-s + 0.603·49-s + 1.54·53-s + 0.269·55-s + 0.605·59-s − 0.103·61-s − 0.615·65-s + 0.645·67-s + 0.474·71-s − 0.422·73-s + 0.455·77-s + 1.36·79-s + 1.28·83-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

\(L(1)\) \(\approx\) \(2.735871869\)
\(L(\frac12)\) \(\approx\) \(2.735871869\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.67T + 5T^{2} \)
7 \( 1 - 3.35T + 7T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 + 2.96T + 13T^{2} \)
17 \( 1 + 0.324T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + 5.92T + 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 - 8.31T + 31T^{2} \)
37 \( 1 + 0.574T + 37T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 + 1.36T + 43T^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 4.64T + 59T^{2} \)
61 \( 1 + 0.806T + 61T^{2} \)
67 \( 1 - 5.28T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 3.61T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−8.086807797974130634410935831725, −7.51021933404629484488238956962, −6.50889013627363891060740454717, −6.03674915500092210776793334278, −5.05889189731661475147995574307, −4.63653767333569513845908612847, −3.75986858880116846713587247271, −2.44188891824670624370552184764, −2.00276263135305488825405961863, −0.892596863415654562334347988198, 0.892596863415654562334347988198, 2.00276263135305488825405961863, 2.44188891824670624370552184764, 3.75986858880116846713587247271, 4.63653767333569513845908612847, 5.05889189731661475147995574307, 6.03674915500092210776793334278, 6.50889013627363891060740454717, 7.51021933404629484488238956962, 8.086807797974130634410935831725

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