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 + 4.25·5-s + 2.10·7-s − 8-s − 4.25·10-s − 4.67·11-s + 0.547·13-s − 2.10·14-s + 16-s − 0.524·17-s − 8.11·19-s + 4.25·20-s + 4.67·22-s + 6.00·23-s + 13.0·25-s − 0.547·26-s + 2.10·28-s + 8.07·29-s + 4.87·31-s − 32-s + 0.524·34-s + 8.94·35-s − 10.1·37-s + 8.11·38-s − 4.25·40-s − 2.58·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.90·5-s + 0.795·7-s − 0.353·8-s − 1.34·10-s − 1.41·11-s + 0.151·13-s − 0.562·14-s + 0.250·16-s − 0.127·17-s − 1.86·19-s + 0.951·20-s + 0.997·22-s + 1.25·23-s + 2.61·25-s − 0.107·26-s + 0.397·28-s + 1.49·29-s + 0.875·31-s − 0.176·32-s + 0.0900·34-s + 1.51·35-s − 1.66·37-s + 1.31·38-s − 0.672·40-s − 0.404·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.267344872$
$L(\frac12)$  $\approx$  $2.267344872$
$L(\frac{3}{2})$   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 \( 1 - 4.25T + 5T^{2} \)
7 \( 1 - 2.10T + 7T^{2} \)
11 \( 1 + 4.67T + 11T^{2} \)
13 \( 1 - 0.547T + 13T^{2} \)
17 \( 1 + 0.524T + 17T^{2} \)
19 \( 1 + 8.11T + 19T^{2} \)
23 \( 1 - 6.00T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 2.58T + 41T^{2} \)
43 \( 1 - 7.72T + 43T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + 0.544T + 53T^{2} \)
59 \( 1 - 7.68T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 - 0.0374T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 4.75T + 79T^{2} \)
83 \( 1 + 7.02T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 - 8.91T + 97T^{2} \)
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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

−8.152393866736439404044304108023, −6.96809799879540151607744424414, −6.57490643296454960794291160433, −5.80474568593609851454156051247, −5.09524482567847079089829300408, −4.65880034204712042903107670677, −3.08080281593125515050954488441, −2.32848667417958022224803693624, −1.88888537209609726301578358831, −0.833309187124058500429288039362, 0.833309187124058500429288039362, 1.88888537209609726301578358831, 2.32848667417958022224803693624, 3.08080281593125515050954488441, 4.65880034204712042903107670677, 5.09524482567847079089829300408, 5.80474568593609851454156051247, 6.57490643296454960794291160433, 6.96809799879540151607744424414, 8.152393866736439404044304108023

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