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  + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s + 4·11-s + 2·12-s − 13-s − 4·16-s + 4·17-s + 2·18-s + 7·19-s + 8·22-s + 6·23-s − 5·25-s − 2·26-s + 27-s − 8·29-s + 31-s − 8·32-s + 4·33-s + 8·34-s + 2·36-s + 7·37-s + 14·38-s − 39-s − 41-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 0.277·13-s − 16-s + 0.970·17-s + 0.471·18-s + 1.60·19-s + 1.70·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 1.41·32-s + 0.696·33-s + 1.37·34-s + 1/3·36-s + 1.15·37-s + 2.27·38-s − 0.160·39-s − 0.156·41-s + ⋯

Functional equation

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

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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.040937367$
$L(\frac12)$  $\approx$  $6.040937367$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.85957688614011467088708385407, −7.21211977446279360622600598854, −6.59580632937904738409413318392, −5.65759312661357432881879911220, −5.23907612885705334959577467821, −4.31592812203653306929802788010, −3.58429756201014766378777015393, −3.20956772096571120049798575246, −2.17540526569732010497996031213, −1.08351983542254918255207166891, 1.08351983542254918255207166891, 2.17540526569732010497996031213, 3.20956772096571120049798575246, 3.58429756201014766378777015393, 4.31592812203653306929802788010, 5.23907612885705334959577467821, 5.65759312661357432881879911220, 6.59580632937904738409413318392, 7.21211977446279360622600598854, 7.85957688614011467088708385407

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