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  + 1.68·2-s + 3-s + 0.832·4-s − 3.40·5-s + 1.68·6-s − 1.96·8-s + 9-s − 5.73·10-s + 1.02·11-s + 0.832·12-s + 3.49·13-s − 3.40·15-s − 4.97·16-s + 0.830·17-s + 1.68·18-s + 1.63·19-s − 2.83·20-s + 1.72·22-s − 1.51·23-s − 1.96·24-s + 6.59·25-s + 5.89·26-s + 27-s − 1.17·29-s − 5.73·30-s − 8.40·31-s − 4.43·32-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.577·3-s + 0.416·4-s − 1.52·5-s + 0.687·6-s − 0.694·8-s + 0.333·9-s − 1.81·10-s + 0.309·11-s + 0.240·12-s + 0.970·13-s − 0.879·15-s − 1.24·16-s + 0.201·17-s + 0.396·18-s + 0.375·19-s − 0.633·20-s + 0.367·22-s − 0.316·23-s − 0.401·24-s + 1.31·25-s + 1.15·26-s + 0.192·27-s − 0.217·29-s − 1.04·30-s − 1.50·31-s − 0.784·32-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 \( 1 - 1.68T + 2T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
11 \( 1 - 1.02T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 0.830T + 17T^{2} \)
19 \( 1 - 1.63T + 19T^{2} \)
23 \( 1 + 1.51T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + 8.40T + 31T^{2} \)
37 \( 1 - 8.14T + 37T^{2} \)
43 \( 1 - 8.80T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 0.417T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + 2.21T + 67T^{2} \)
71 \( 1 + 1.00T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 9.00T + 79T^{2} \)
83 \( 1 + 1.44T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
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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

−7.73053299774155446354678564659, −7.03075452089457877988913061864, −6.16250167471392752593322466842, −5.49313559147018267645038337392, −4.34241650694689218618819818747, −4.20427793304421142235675427069, −3.35499289728178913107001226196, −2.92894829438886128393629113843, −1.46786501514113450902943218355, 0, 1.46786501514113450902943218355, 2.92894829438886128393629113843, 3.35499289728178913107001226196, 4.20427793304421142235675427069, 4.34241650694689218618819818747, 5.49313559147018267645038337392, 6.16250167471392752593322466842, 7.03075452089457877988913061864, 7.73053299774155446354678564659

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