Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.614·5-s − 3.46·7-s − 3.80·11-s − 2.93·13-s − 3.15·17-s − 5.58·19-s + 0.574·23-s − 4.62·25-s + 3.25·29-s − 1.74·31-s − 2.12·35-s − 4.35·37-s + 1.51·41-s + 9.97·43-s + 8.29·47-s + 4.99·49-s − 0.465·53-s − 2.34·55-s + 9.58·59-s − 6.22·61-s − 1.80·65-s + 3.41·67-s + 8.58·71-s − 9.90·73-s + 13.1·77-s + 10.2·79-s + 8.39·83-s + ⋯
L(s)  = 1  + 0.275·5-s − 1.30·7-s − 1.14·11-s − 0.812·13-s − 0.765·17-s − 1.28·19-s + 0.119·23-s − 0.924·25-s + 0.603·29-s − 0.313·31-s − 0.359·35-s − 0.716·37-s + 0.236·41-s + 1.52·43-s + 1.20·47-s + 0.713·49-s − 0.0638·53-s − 0.315·55-s + 1.24·59-s − 0.797·61-s − 0.223·65-s + 0.416·67-s + 1.01·71-s − 1.15·73-s + 1.50·77-s + 1.15·79-s + 0.921·83-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

\( d \)  =  \(2\)
\( N \)  =  \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6012,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.7690382344\)
\(L(\frac12)\)  \(\approx\)  \(0.7690382344\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 0.614T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 3.80T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 3.15T + 17T^{2} \)
19 \( 1 + 5.58T + 19T^{2} \)
23 \( 1 - 0.574T + 23T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + 1.74T + 31T^{2} \)
37 \( 1 + 4.35T + 37T^{2} \)
41 \( 1 - 1.51T + 41T^{2} \)
43 \( 1 - 9.97T + 43T^{2} \)
47 \( 1 - 8.29T + 47T^{2} \)
53 \( 1 + 0.465T + 53T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 + 6.22T + 61T^{2} \)
67 \( 1 - 3.41T + 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 + 9.90T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 8.39T + 83T^{2} \)
89 \( 1 - 7.12T + 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.015691554841912529144819243680, −7.32710114399640974803504918691, −6.61464871371783242190414027291, −6.02407451502591836181516925387, −5.28196737942227455087274558946, −4.42759416258627250006456211627, −3.62510106029962839840631464629, −2.58105470431620863279746074866, −2.20414976932895680863978180055, −0.42439600878059316446815568226, 0.42439600878059316446815568226, 2.20414976932895680863978180055, 2.58105470431620863279746074866, 3.62510106029962839840631464629, 4.42759416258627250006456211627, 5.28196737942227455087274558946, 6.02407451502591836181516925387, 6.61464871371783242190414027291, 7.32710114399640974803504918691, 8.015691554841912529144819243680

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