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  − 2.37·2-s + 3-s + 3.66·4-s + 3.68·5-s − 2.37·6-s − 3.95·8-s + 9-s − 8.77·10-s − 6.26·11-s + 3.66·12-s − 3.25·13-s + 3.68·15-s + 2.09·16-s + 0.146·17-s − 2.37·18-s − 1.78·19-s + 13.5·20-s + 14.9·22-s − 3.11·23-s − 3.95·24-s + 8.59·25-s + 7.74·26-s + 27-s + 4.08·29-s − 8.77·30-s + 1.31·31-s + 2.93·32-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.577·3-s + 1.83·4-s + 1.64·5-s − 0.971·6-s − 1.39·8-s + 0.333·9-s − 2.77·10-s − 1.88·11-s + 1.05·12-s − 0.903·13-s + 0.952·15-s + 0.523·16-s + 0.0354·17-s − 0.560·18-s − 0.409·19-s + 3.02·20-s + 3.17·22-s − 0.649·23-s − 0.808·24-s + 1.71·25-s + 1.51·26-s + 0.192·27-s + 0.758·29-s − 1.60·30-s + 0.236·31-s + 0.518·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.175599714$
$L(\frac12)$  $\approx$  $1.175599714$
$L(\frac{3}{2})$   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 \( 1 + 2.37T + 2T^{2} \)
5 \( 1 - 3.68T + 5T^{2} \)
11 \( 1 + 6.26T + 11T^{2} \)
13 \( 1 + 3.25T + 13T^{2} \)
17 \( 1 - 0.146T + 17T^{2} \)
19 \( 1 + 1.78T + 19T^{2} \)
23 \( 1 + 3.11T + 23T^{2} \)
29 \( 1 - 4.08T + 29T^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 4.41T + 53T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 - 6.02T + 61T^{2} \)
67 \( 1 - 1.38T + 67T^{2} \)
71 \( 1 - 8.92T + 71T^{2} \)
73 \( 1 - 8.93T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 1.75T + 89T^{2} \)
97 \( 1 - 5.53T + 97T^{2} \)
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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

−8.307400002723638075158505843008, −7.55935357859713209641704784641, −6.97617156778355943184561651509, −6.14770167727094497992088105721, −5.40653520402656686234272744514, −4.63062080905720398584531848284, −2.98214531134097341498138386115, −2.28828830787101612062621498929, −2.01306827257345689423730530539, −0.68413019504376258509266011756, 0.68413019504376258509266011756, 2.01306827257345689423730530539, 2.28828830787101612062621498929, 2.98214531134097341498138386115, 4.63062080905720398584531848284, 5.40653520402656686234272744514, 6.14770167727094497992088105721, 6.97617156778355943184561651509, 7.55935357859713209641704784641, 8.307400002723638075158505843008

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