Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.404·2-s + 3-s − 1.83·4-s − 3.62·5-s + 0.404·6-s − 1.55·8-s + 9-s − 1.46·10-s − 5.64·11-s − 1.83·12-s + 4.03·13-s − 3.62·15-s + 3.04·16-s + 2.00·17-s + 0.404·18-s − 1.50·19-s + 6.66·20-s − 2.28·22-s + 7.05·23-s − 1.55·24-s + 8.15·25-s + 1.63·26-s + 27-s + 5.14·29-s − 1.46·30-s − 3.25·31-s + 4.33·32-s + ⋯
L(s)  = 1  + 0.285·2-s + 0.577·3-s − 0.918·4-s − 1.62·5-s + 0.165·6-s − 0.548·8-s + 0.333·9-s − 0.463·10-s − 1.70·11-s − 0.530·12-s + 1.11·13-s − 0.936·15-s + 0.761·16-s + 0.487·17-s + 0.0952·18-s − 0.346·19-s + 1.48·20-s − 0.486·22-s + 1.47·23-s − 0.316·24-s + 1.63·25-s + 0.320·26-s + 0.192·27-s + 0.955·29-s − 0.267·30-s − 0.585·31-s + 0.766·32-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 0.404T + 2T^{2} \)
5 \( 1 + 3.62T + 5T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 - 4.03T + 13T^{2} \)
17 \( 1 - 2.00T + 17T^{2} \)
19 \( 1 + 1.50T + 19T^{2} \)
23 \( 1 - 7.05T + 23T^{2} \)
29 \( 1 - 5.14T + 29T^{2} \)
31 \( 1 + 3.25T + 31T^{2} \)
37 \( 1 - 0.612T + 37T^{2} \)
43 \( 1 - 1.57T + 43T^{2} \)
47 \( 1 - 0.256T + 47T^{2} \)
53 \( 1 + 7.87T + 53T^{2} \)
59 \( 1 + 4.32T + 59T^{2} \)
61 \( 1 - 2.81T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 + 1.04T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 9.74T + 89T^{2} \)
97 \( 1 + 8.22T + 97T^{2} \)
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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

−7.81921665968201410849077332604, −7.36254527646386470595382043208, −6.28858005106661484241046341326, −5.27371661599342531654202982638, −4.72280978962782935669008917864, −4.00147682602157818230121582838, −3.29620204775311241767382461047, −2.80453662260883112862395836979, −1.07436180705113325146750979872, 0, 1.07436180705113325146750979872, 2.80453662260883112862395836979, 3.29620204775311241767382461047, 4.00147682602157818230121582838, 4.72280978962782935669008917864, 5.27371661599342531654202982638, 6.28858005106661484241046341326, 7.36254527646386470595382043208, 7.81921665968201410849077332604

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