Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
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 − 3-s − 4-s − 3·5-s + 6-s + 3·8-s + 9-s + 3·10-s − 6·11-s + 12-s + 7·13-s + 3·15-s − 16-s + 6·17-s − 18-s + 6·19-s + 3·20-s + 6·22-s + 5·23-s − 3·24-s + 4·25-s − 7·26-s − 27-s + 3·29-s − 3·30-s − 5·32-s + 6·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.948·10-s − 1.80·11-s + 0.288·12-s + 1.94·13-s + 0.774·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 1.27·22-s + 1.04·23-s − 0.612·24-s + 4/5·25-s − 1.37·26-s − 0.192·27-s + 0.557·29-s − 0.547·30-s − 0.883·32-s + 1.04·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7276301775$
$L(\frac12)$  $\approx$  $0.7276301775$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.052626830535662893221176015466, −7.63602410646239790739546115596, −6.99184855984182345692502648899, −5.70692396271790037561039177754, −5.30628630371854907688301440175, −4.46249568554153281294726319283, −3.63186048390990226223884600245, −3.00981877031938564956438485856, −1.21442536501331329439792892628, −0.62958694301807171385699555822, 0.62958694301807171385699555822, 1.21442536501331329439792892628, 3.00981877031938564956438485856, 3.63186048390990226223884600245, 4.46249568554153281294726319283, 5.30628630371854907688301440175, 5.70692396271790037561039177754, 6.99184855984182345692502648899, 7.63602410646239790739546115596, 8.052626830535662893221176015466

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