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 − 1.12·5-s − 0.686·7-s − 8-s + 1.12·10-s + 3.05·11-s − 3.99·13-s + 0.686·14-s + 16-s + 5.80·17-s + 8.32·19-s − 1.12·20-s − 3.05·22-s − 2.20·23-s − 3.73·25-s + 3.99·26-s − 0.686·28-s + 6.02·29-s + 8.16·31-s − 32-s − 5.80·34-s + 0.772·35-s + 4.59·37-s − 8.32·38-s + 1.12·40-s + 5.82·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.503·5-s − 0.259·7-s − 0.353·8-s + 0.355·10-s + 0.920·11-s − 1.10·13-s + 0.183·14-s + 0.250·16-s + 1.40·17-s + 1.90·19-s − 0.251·20-s − 0.651·22-s − 0.459·23-s − 0.746·25-s + 0.782·26-s − 0.129·28-s + 1.11·29-s + 1.46·31-s − 0.176·32-s − 0.994·34-s + 0.130·35-s + 0.754·37-s − 1.34·38-s + 0.177·40-s + 0.910·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.363794449$
$L(\frac12)$  $\approx$  $1.363794449$
$L(\frac{3}{2})$   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 \( 1 + 1.12T + 5T^{2} \)
7 \( 1 + 0.686T + 7T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 + 3.99T + 13T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 - 8.32T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 - 6.02T + 29T^{2} \)
31 \( 1 - 8.16T + 31T^{2} \)
37 \( 1 - 4.59T + 37T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 - 0.944T + 43T^{2} \)
47 \( 1 + 6.19T + 47T^{2} \)
53 \( 1 + 8.18T + 53T^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 6.33T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 3.33T + 73T^{2} \)
79 \( 1 + 8.49T + 79T^{2} \)
83 \( 1 - 6.34T + 83T^{2} \)
89 \( 1 - 6.07T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.85074920079866069107167402261, −7.36188626042468557205260888189, −6.57066902072783283493694675763, −5.91137927951198106330319943778, −5.05575752875540403641433391988, −4.24839671283980084333966905542, −3.28063526915871358777355638133, −2.78521047784654782706156700269, −1.48481388395072326007553031886, −0.69125529516605066899667742817, 0.69125529516605066899667742817, 1.48481388395072326007553031886, 2.78521047784654782706156700269, 3.28063526915871358777355638133, 4.24839671283980084333966905542, 5.05575752875540403641433391988, 5.91137927951198106330319943778, 6.57066902072783283493694675763, 7.36188626042468557205260888189, 7.85074920079866069107167402261

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