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 + 1.72·5-s − 2.60·7-s + 8-s + 1.72·10-s − 5.87·11-s + 3.97·13-s − 2.60·14-s + 16-s + 3.69·17-s − 0.972·19-s + 1.72·20-s − 5.87·22-s − 1.45·23-s − 2.02·25-s + 3.97·26-s − 2.60·28-s + 0.793·29-s − 5.90·31-s + 32-s + 3.69·34-s − 4.49·35-s − 1.91·37-s − 0.972·38-s + 1.72·40-s + 2.77·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.771·5-s − 0.984·7-s + 0.353·8-s + 0.545·10-s − 1.77·11-s + 1.10·13-s − 0.696·14-s + 0.250·16-s + 0.894·17-s − 0.223·19-s + 0.385·20-s − 1.25·22-s − 0.302·23-s − 0.404·25-s + 0.779·26-s − 0.492·28-s + 0.147·29-s − 1.06·31-s + 0.176·32-s + 0.632·34-s − 0.759·35-s − 0.314·37-s − 0.157·38-s + 0.272·40-s + 0.434·41-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 \( 1 - 1.72T + 5T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 5.87T + 11T^{2} \)
13 \( 1 - 3.97T + 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 + 0.972T + 19T^{2} \)
23 \( 1 + 1.45T + 23T^{2} \)
29 \( 1 - 0.793T + 29T^{2} \)
31 \( 1 + 5.90T + 31T^{2} \)
37 \( 1 + 1.91T + 37T^{2} \)
41 \( 1 - 2.77T + 41T^{2} \)
43 \( 1 - 0.964T + 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 1.69T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 - 1.13T + 67T^{2} \)
71 \( 1 + 3.26T + 71T^{2} \)
73 \( 1 - 3.64T + 73T^{2} \)
79 \( 1 + 6.18T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 2.08T + 89T^{2} \)
97 \( 1 - 4.60T + 97T^{2} \)
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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

−7.51391630321469010795255334002, −6.46569175820769368957518210720, −6.04659513592741483519610030788, −5.47884244358887686446545055227, −4.83785527167539463032988493053, −3.71710371978438042404892417336, −3.18047784287463671012766992736, −2.41680377049469375870152238854, −1.50336970869346827549236182273, 0, 1.50336970869346827549236182273, 2.41680377049469375870152238854, 3.18047784287463671012766992736, 3.71710371978438042404892417336, 4.83785527167539463032988493053, 5.47884244358887686446545055227, 6.04659513592741483519610030788, 6.46569175820769368957518210720, 7.51391630321469010795255334002

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