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 + 1.06·5-s + 2.41·7-s − 8-s − 1.06·10-s + 2.87·11-s + 6.31·13-s − 2.41·14-s + 16-s + 5.80·17-s + 2.51·19-s + 1.06·20-s − 2.87·22-s + 4.87·23-s − 3.86·25-s − 6.31·26-s + 2.41·28-s + 5.18·29-s − 7.91·31-s − 32-s − 5.80·34-s + 2.57·35-s − 4.64·37-s − 2.51·38-s − 1.06·40-s + 10.6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.477·5-s + 0.912·7-s − 0.353·8-s − 0.337·10-s + 0.867·11-s + 1.75·13-s − 0.644·14-s + 0.250·16-s + 1.40·17-s + 0.577·19-s + 0.238·20-s − 0.613·22-s + 1.01·23-s − 0.772·25-s − 1.23·26-s + 0.456·28-s + 0.962·29-s − 1.42·31-s − 0.176·32-s − 0.995·34-s + 0.435·35-s − 0.763·37-s − 0.408·38-s − 0.168·40-s + 1.66·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.629177526$
$L(\frac12)$  $\approx$  $2.629177526$
$L(\frac{3}{2})$   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 \( 1 - 1.06T + 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 - 6.31T + 13T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 - 5.18T + 29T^{2} \)
31 \( 1 + 7.91T + 31T^{2} \)
37 \( 1 + 4.64T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 + 6.78T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 3.88T + 71T^{2} \)
73 \( 1 + 1.97T + 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 6.69T + 97T^{2} \)
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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

−7.78454567856038064743862341856, −7.38829353704707657793698966131, −6.43230303838675870174565818241, −5.79704511381062995291620105428, −5.31373658088635829192491670705, −4.11719281944565495443849491682, −3.51710140571995602602576659086, −2.49353153984483218173765013266, −1.24340086217206695307026680668, −1.20847776283533113375550546205, 1.20847776283533113375550546205, 1.24340086217206695307026680668, 2.49353153984483218173765013266, 3.51710140571995602602576659086, 4.11719281944565495443849491682, 5.31373658088635829192491670705, 5.79704511381062995291620105428, 6.43230303838675870174565818241, 7.38829353704707657793698966131, 7.78454567856038064743862341856

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