Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
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-s + 4-s − 2.26·5-s − 0.397·7-s − 8-s + 2.26·10-s + 3.96·11-s + 0.823·13-s + 0.397·14-s + 16-s − 6.85·17-s − 8.38·19-s − 2.26·20-s − 3.96·22-s + 8.28·23-s + 0.142·25-s − 0.823·26-s − 0.397·28-s + 8.78·29-s + 4.00·31-s − 32-s + 6.85·34-s + 0.902·35-s + 0.694·37-s + 8.38·38-s + 2.26·40-s + 9.11·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.01·5-s − 0.150·7-s − 0.353·8-s + 0.717·10-s + 1.19·11-s + 0.228·13-s + 0.106·14-s + 0.250·16-s − 1.66·17-s − 1.92·19-s − 0.507·20-s − 0.846·22-s + 1.72·23-s + 0.0284·25-s − 0.161·26-s − 0.0751·28-s + 1.63·29-s + 0.719·31-s − 0.176·32-s + 1.17·34-s + 0.152·35-s + 0.114·37-s + 1.36·38-s + 0.358·40-s + 1.42·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8046} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8620038322$
$L(\frac12)$  $\approx$  $0.8620038322$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;149\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
149 \( 1 - T \)
good5 \( 1 + 2.26T + 5T^{2} \)
7 \( 1 + 0.397T + 7T^{2} \)
11 \( 1 - 3.96T + 11T^{2} \)
13 \( 1 - 0.823T + 13T^{2} \)
17 \( 1 + 6.85T + 17T^{2} \)
19 \( 1 + 8.38T + 19T^{2} \)
23 \( 1 - 8.28T + 23T^{2} \)
29 \( 1 - 8.78T + 29T^{2} \)
31 \( 1 - 4.00T + 31T^{2} \)
37 \( 1 - 0.694T + 37T^{2} \)
41 \( 1 - 9.11T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + 7.80T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 8.40T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 6.27T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 - 8.76T + 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 8.05T + 97T^{2} \)
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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

−8.096678252610705263237505235399, −7.05656403793048832062720605146, −6.50679236300598583076816561127, −6.27386213145412727571797457639, −4.62171252912898233233221690437, −4.45234736474307324411243657620, −3.45667233954838104650042714194, −2.62781327873148395839399223149, −1.60087766238926537706458266803, −0.51864563540575480112053900073, 0.51864563540575480112053900073, 1.60087766238926537706458266803, 2.62781327873148395839399223149, 3.45667233954838104650042714194, 4.45234736474307324411243657620, 4.62171252912898233233221690437, 6.27386213145412727571797457639, 6.50679236300598583076816561127, 7.05656403793048832062720605146, 8.096678252610705263237505235399

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