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 + 1.09·5-s − 2.72·7-s − 8-s − 1.09·10-s − 4.35·11-s − 3.83·13-s + 2.72·14-s + 16-s − 4.52·17-s − 1.46·19-s + 1.09·20-s + 4.35·22-s − 2.47·23-s − 3.79·25-s + 3.83·26-s − 2.72·28-s + 9.77·29-s − 0.816·31-s − 32-s + 4.52·34-s − 2.99·35-s + 3.29·37-s + 1.46·38-s − 1.09·40-s − 6.03·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.490·5-s − 1.03·7-s − 0.353·8-s − 0.346·10-s − 1.31·11-s − 1.06·13-s + 0.729·14-s + 0.250·16-s − 1.09·17-s − 0.335·19-s + 0.245·20-s + 0.929·22-s − 0.517·23-s − 0.759·25-s + 0.752·26-s − 0.515·28-s + 1.81·29-s − 0.146·31-s − 0.176·32-s + 0.775·34-s − 0.505·35-s + 0.542·37-s + 0.237·38-s − 0.173·40-s − 0.942·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4396941619$
$L(\frac12)$  $\approx$  $0.4396941619$
$L(\frac{3}{2})$   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 \( 1 - 1.09T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + 3.83T + 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + 1.46T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 9.77T + 29T^{2} \)
31 \( 1 + 0.816T + 31T^{2} \)
37 \( 1 - 3.29T + 37T^{2} \)
41 \( 1 + 6.03T + 41T^{2} \)
43 \( 1 - 2.88T + 43T^{2} \)
47 \( 1 + 1.80T + 47T^{2} \)
53 \( 1 - 9.70T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 9.19T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 9.35T + 73T^{2} \)
79 \( 1 - 6.85T + 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 6.62T + 97T^{2} \)
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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

−7.71051129216511671164575362629, −7.33821378565279071781152827315, −6.27798405473050133635141232466, −6.16292383446923167743503995247, −5.04128388424012822620026697901, −4.40957219262180012473086552562, −3.13505107772201430650574322461, −2.58573154455468588403277514170, −1.87973679816920694651449836737, −0.33800035764872442640047898198, 0.33800035764872442640047898198, 1.87973679816920694651449836737, 2.58573154455468588403277514170, 3.13505107772201430650574322461, 4.40957219262180012473086552562, 5.04128388424012822620026697901, 6.16292383446923167743503995247, 6.27798405473050133635141232466, 7.33821378565279071781152827315, 7.71051129216511671164575362629

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