Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.96·5-s − 4.72·7-s + 8-s + 1.96·10-s + 2.36·11-s − 0.742·13-s − 4.72·14-s + 16-s − 1.40·17-s − 2.69·19-s + 1.96·20-s + 2.36·22-s − 3.63·23-s − 1.13·25-s − 0.742·26-s − 4.72·28-s + 0.980·29-s + 3.58·31-s + 32-s − 1.40·34-s − 9.28·35-s + 10.6·37-s − 2.69·38-s + 1.96·40-s − 6.88·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.879·5-s − 1.78·7-s + 0.353·8-s + 0.621·10-s + 0.713·11-s − 0.205·13-s − 1.26·14-s + 0.250·16-s − 0.339·17-s − 0.617·19-s + 0.439·20-s + 0.504·22-s − 0.758·23-s − 0.226·25-s − 0.145·26-s − 0.892·28-s + 0.182·29-s + 0.644·31-s + 0.176·32-s − 0.240·34-s − 1.56·35-s + 1.74·37-s − 0.436·38-s + 0.310·40-s − 1.07·41-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  =  1
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
149 \( 1 + T \)
good5 \( 1 - 1.96T + 5T^{2} \)
7 \( 1 + 4.72T + 7T^{2} \)
11 \( 1 - 2.36T + 11T^{2} \)
13 \( 1 + 0.742T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + 2.69T + 19T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 - 0.980T + 29T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 + 1.80T + 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 - 1.45T + 53T^{2} \)
59 \( 1 + 2.49T + 59T^{2} \)
61 \( 1 + 7.04T + 61T^{2} \)
67 \( 1 + 4.95T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 7.71T + 73T^{2} \)
79 \( 1 - 2.59T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 1.95T + 89T^{2} \)
97 \( 1 - 0.529T + 97T^{2} \)
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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

−7.16897684495261393879638530404, −6.45140018709911269966767666911, −6.21238250181044560891387910514, −5.63197974829848806704426319584, −4.52563732665533942033413602418, −3.94571394372593372809778955046, −3.05816765419673316044942202459, −2.47877497715247089093561173255, −1.47890566537837333370781901050, 0, 1.47890566537837333370781901050, 2.47877497715247089093561173255, 3.05816765419673316044942202459, 3.94571394372593372809778955046, 4.52563732665533942033413602418, 5.63197974829848806704426319584, 6.21238250181044560891387910514, 6.45140018709911269966767666911, 7.16897684495261393879638530404

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