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 + 2.71·5-s + 1.67·7-s + 8-s + 2.71·10-s − 4.37·11-s − 2.34·13-s + 1.67·14-s + 16-s − 4.09·17-s − 7.26·19-s + 2.71·20-s − 4.37·22-s + 0.0981·23-s + 2.34·25-s − 2.34·26-s + 1.67·28-s − 5.89·29-s − 1.88·31-s + 32-s − 4.09·34-s + 4.54·35-s − 3.13·37-s − 7.26·38-s + 2.71·40-s − 0.419·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.21·5-s + 0.634·7-s + 0.353·8-s + 0.857·10-s − 1.31·11-s − 0.651·13-s + 0.448·14-s + 0.250·16-s − 0.993·17-s − 1.66·19-s + 0.606·20-s − 0.932·22-s + 0.0204·23-s + 0.469·25-s − 0.460·26-s + 0.317·28-s − 1.09·29-s − 0.337·31-s + 0.176·32-s − 0.702·34-s + 0.768·35-s − 0.515·37-s − 1.17·38-s + 0.428·40-s − 0.0655·41-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 \( 1 - 2.71T + 5T^{2} \)
7 \( 1 - 1.67T + 7T^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 + 4.09T + 17T^{2} \)
19 \( 1 + 7.26T + 19T^{2} \)
23 \( 1 - 0.0981T + 23T^{2} \)
29 \( 1 + 5.89T + 29T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + 3.13T + 37T^{2} \)
41 \( 1 + 0.419T + 41T^{2} \)
43 \( 1 + 0.0900T + 43T^{2} \)
47 \( 1 + 1.12T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 7.63T + 59T^{2} \)
61 \( 1 - 3.77T + 61T^{2} \)
67 \( 1 - 3.12T + 67T^{2} \)
71 \( 1 + 3.20T + 71T^{2} \)
73 \( 1 - 1.15T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 7.11T + 89T^{2} \)
97 \( 1 + 4.78T + 97T^{2} \)
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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

−7.29010834833294823562817062399, −6.73932884752461415458606063284, −5.83708383514673372724364650657, −5.46882120172705805809124504378, −4.72412222035164578696921472611, −4.14027192554389927723742519664, −2.90933134584048003152153981378, −2.16123111618924675425393508721, −1.81814527069704662786472965717, 0, 1.81814527069704662786472965717, 2.16123111618924675425393508721, 2.90933134584048003152153981378, 4.14027192554389927723742519664, 4.72412222035164578696921472611, 5.46882120172705805809124504378, 5.83708383514673372724364650657, 6.73932884752461415458606063284, 7.29010834833294823562817062399

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