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  + 1.95·2-s − 3-s + 1.80·4-s − 2.46·5-s − 1.95·6-s − 0.380·8-s + 9-s − 4.81·10-s + 3.40·11-s − 1.80·12-s − 3.78·13-s + 2.46·15-s − 4.35·16-s + 7.48·17-s + 1.95·18-s − 1.93·19-s − 4.45·20-s + 6.65·22-s − 1.38·23-s + 0.380·24-s + 1.08·25-s − 7.38·26-s − 27-s + 4.33·29-s + 4.81·30-s − 10.1·31-s − 7.72·32-s + ⋯
L(s)  = 1  + 1.37·2-s − 0.577·3-s + 0.902·4-s − 1.10·5-s − 0.796·6-s − 0.134·8-s + 0.333·9-s − 1.52·10-s + 1.02·11-s − 0.521·12-s − 1.05·13-s + 0.636·15-s − 1.08·16-s + 1.81·17-s + 0.459·18-s − 0.443·19-s − 0.995·20-s + 1.41·22-s − 0.289·23-s + 0.0776·24-s + 0.217·25-s − 1.44·26-s − 0.192·27-s + 0.804·29-s + 0.878·30-s − 1.82·31-s − 1.36·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.281784147$
$L(\frac12)$  $\approx$  $2.281784147$
$L(\frac{3}{2})$   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 \( 1 - 1.95T + 2T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
11 \( 1 - 3.40T + 11T^{2} \)
13 \( 1 + 3.78T + 13T^{2} \)
17 \( 1 - 7.48T + 17T^{2} \)
19 \( 1 + 1.93T + 19T^{2} \)
23 \( 1 + 1.38T + 23T^{2} \)
29 \( 1 - 4.33T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 8.77T + 37T^{2} \)
43 \( 1 + 5.71T + 43T^{2} \)
47 \( 1 + 4.34T + 47T^{2} \)
53 \( 1 - 4.49T + 53T^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 8.88T + 67T^{2} \)
71 \( 1 - 16.4T + 71T^{2} \)
73 \( 1 - 0.523T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 18.8T + 89T^{2} \)
97 \( 1 - 8.05T + 97T^{2} \)
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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

−7.75701887156819899425071337920, −7.23852069985989579988852146966, −6.46506789037705813561181902928, −5.79337577576862082648779813570, −5.07252695170313081784099952713, −4.46366525854824016147443027122, −3.73031253153386557046897972961, −3.30349889100516836714125317314, −2.07229734271217712680397666182, −0.65106801890509515454723424798, 0.65106801890509515454723424798, 2.07229734271217712680397666182, 3.30349889100516836714125317314, 3.73031253153386557046897972961, 4.46366525854824016147443027122, 5.07252695170313081784099952713, 5.79337577576862082648779813570, 6.46506789037705813561181902928, 7.23852069985989579988852146966, 7.75701887156819899425071337920

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