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 − 1.42·7-s − 2.19·11-s + 1.74·13-s − 3.26·17-s + 5.89·19-s − 6.12·23-s − 25-s + 0.321·29-s + 1.75·31-s + 2.85·35-s − 2.47·37-s + 5.98·41-s − 3.49·43-s − 2.39·47-s − 4.96·49-s + 1.51·53-s + 4.38·55-s − 5.65·59-s − 4.51·61-s − 3.49·65-s − 9.61·67-s + 6.66·71-s + 2.86·73-s + 3.12·77-s + 10.4·79-s + 7.24·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.539·7-s − 0.660·11-s + 0.484·13-s − 0.791·17-s + 1.35·19-s − 1.27·23-s − 0.200·25-s + 0.0596·29-s + 0.314·31-s + 0.482·35-s − 0.406·37-s + 0.935·41-s − 0.533·43-s − 0.349·47-s − 0.709·49-s + 0.208·53-s + 0.590·55-s − 0.736·59-s − 0.578·61-s − 0.433·65-s − 1.17·67-s + 0.791·71-s + 0.335·73-s + 0.356·77-s + 1.18·79-s + 0.795·83-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

\(L(1)\) \(\approx\) \(0.9666093334\)
\(L(\frac12)\) \(\approx\) \(0.9666093334\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.42T + 7T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 - 1.74T + 13T^{2} \)
17 \( 1 + 3.26T + 17T^{2} \)
19 \( 1 - 5.89T + 19T^{2} \)
23 \( 1 + 6.12T + 23T^{2} \)
29 \( 1 - 0.321T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 - 5.98T + 41T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 + 2.39T + 47T^{2} \)
53 \( 1 - 1.51T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 4.51T + 61T^{2} \)
67 \( 1 + 9.61T + 67T^{2} \)
71 \( 1 - 6.66T + 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 7.24T + 83T^{2} \)
89 \( 1 - 0.256T + 89T^{2} \)
97 \( 1 - 5.83T + 97T^{2} \)
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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

−7.85263500611444156433164145188, −7.63633815124245454301925394849, −6.63599822318493047158011954518, −6.03706478825733991535153275349, −5.16811304268384140485941288550, −4.36207095646761771736990946529, −3.60531847394915253201406939915, −2.96697834252916844500366018324, −1.87299525056923771493175416382, −0.50084900444925075292918399694, 0.50084900444925075292918399694, 1.87299525056923771493175416382, 2.96697834252916844500366018324, 3.60531847394915253201406939915, 4.36207095646761771736990946529, 5.16811304268384140485941288550, 6.03706478825733991535153275349, 6.63599822318493047158011954518, 7.63633815124245454301925394849, 7.85263500611444156433164145188

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