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  + 1.49·2-s + 3-s + 0.236·4-s + 1.30·5-s + 1.49·6-s − 2.63·8-s + 9-s + 1.94·10-s − 0.750·11-s + 0.236·12-s − 4.40·13-s + 1.30·15-s − 4.41·16-s − 4.00·17-s + 1.49·18-s + 5.38·19-s + 0.308·20-s − 1.12·22-s + 3.23·23-s − 2.63·24-s − 3.30·25-s − 6.58·26-s + 27-s − 3.07·29-s + 1.94·30-s − 9.02·31-s − 1.33·32-s + ⋯
L(s)  = 1  + 1.05·2-s + 0.577·3-s + 0.118·4-s + 0.581·5-s + 0.610·6-s − 0.932·8-s + 0.333·9-s + 0.615·10-s − 0.226·11-s + 0.0683·12-s − 1.22·13-s + 0.335·15-s − 1.10·16-s − 0.970·17-s + 0.352·18-s + 1.23·19-s + 0.0688·20-s − 0.239·22-s + 0.674·23-s − 0.538·24-s − 0.661·25-s − 1.29·26-s + 0.192·27-s − 0.571·29-s + 0.355·30-s − 1.62·31-s − 0.235·32-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 \( 1 - 1.49T + 2T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
11 \( 1 + 0.750T + 11T^{2} \)
13 \( 1 + 4.40T + 13T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 + 3.07T + 29T^{2} \)
31 \( 1 + 9.02T + 31T^{2} \)
37 \( 1 - 0.222T + 37T^{2} \)
43 \( 1 - 0.320T + 43T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 0.199T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 9.59T + 83T^{2} \)
89 \( 1 - 4.12T + 89T^{2} \)
97 \( 1 + 6.19T + 97T^{2} \)
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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

−7.45637783840869561094447684338, −7.09026659879390991628276508765, −6.05619179646713583675793000794, −5.42077931935798583326990319040, −4.84579644307460114277933228166, −4.09934883975286769602376489277, −3.23078949462399099190246863833, −2.59204270470538141261233607917, −1.73069096678099872152357507310, 0, 1.73069096678099872152357507310, 2.59204270470538141261233607917, 3.23078949462399099190246863833, 4.09934883975286769602376489277, 4.84579644307460114277933228166, 5.42077931935798583326990319040, 6.05619179646713583675793000794, 7.09026659879390991628276508765, 7.45637783840869561094447684338

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