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 − 1.17·5-s + 0.506·7-s − 8-s + 1.17·10-s − 3.88·11-s − 3.61·13-s − 0.506·14-s + 16-s + 6.95·17-s − 2.33·19-s − 1.17·20-s + 3.88·22-s − 1.72·23-s − 3.62·25-s + 3.61·26-s + 0.506·28-s + 4.21·29-s − 3.29·31-s − 32-s − 6.95·34-s − 0.592·35-s − 7.04·37-s + 2.33·38-s + 1.17·40-s + 10.5·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.523·5-s + 0.191·7-s − 0.353·8-s + 0.370·10-s − 1.17·11-s − 1.00·13-s − 0.135·14-s + 0.250·16-s + 1.68·17-s − 0.536·19-s − 0.261·20-s + 0.827·22-s − 0.359·23-s − 0.725·25-s + 0.708·26-s + 0.0957·28-s + 0.783·29-s − 0.591·31-s − 0.176·32-s − 1.19·34-s − 0.100·35-s − 1.15·37-s + 0.379·38-s + 0.185·40-s + 1.64·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7293389437$
$L(\frac12)$  $\approx$  $0.7293389437$
$L(\frac{3}{2})$   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 \( 1 + 1.17T + 5T^{2} \)
7 \( 1 - 0.506T + 7T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + 3.61T + 13T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 + 2.33T + 19T^{2} \)
23 \( 1 + 1.72T + 23T^{2} \)
29 \( 1 - 4.21T + 29T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 6.84T + 43T^{2} \)
47 \( 1 + 4.17T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 7.05T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 5.00T + 67T^{2} \)
71 \( 1 + 0.923T + 71T^{2} \)
73 \( 1 - 2.46T + 73T^{2} \)
79 \( 1 + 7.80T + 79T^{2} \)
83 \( 1 + 8.00T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 1.02T + 97T^{2} \)
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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

−7.87404503084185260628619242353, −7.42383739945638428496605581322, −6.66879578724653918465213848597, −5.64564112992254072604169051498, −5.21472088478674009718901702141, −4.26377024393714353493166753318, −3.33974791175690850052360831926, −2.59851873277657390121905435149, −1.71345715946999462378600293517, −0.45838413417870085481505307293, 0.45838413417870085481505307293, 1.71345715946999462378600293517, 2.59851873277657390121905435149, 3.33974791175690850052360831926, 4.26377024393714353493166753318, 5.21472088478674009718901702141, 5.64564112992254072604169051498, 6.66879578724653918465213848597, 7.42383739945638428496605581322, 7.87404503084185260628619242353

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