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  + 2.26·2-s + 3-s + 3.11·4-s + 3.51·5-s + 2.26·6-s + 2.53·8-s + 9-s + 7.95·10-s − 3.79·11-s + 3.11·12-s + 1.46·13-s + 3.51·15-s − 0.508·16-s + 6.67·17-s + 2.26·18-s − 2.27·19-s + 10.9·20-s − 8.58·22-s + 8.49·23-s + 2.53·24-s + 7.35·25-s + 3.30·26-s + 27-s − 5.04·29-s + 7.95·30-s − 1.44·31-s − 6.21·32-s + ⋯
L(s)  = 1  + 1.59·2-s + 0.577·3-s + 1.55·4-s + 1.57·5-s + 0.923·6-s + 0.895·8-s + 0.333·9-s + 2.51·10-s − 1.14·11-s + 0.900·12-s + 0.405·13-s + 0.907·15-s − 0.127·16-s + 1.61·17-s + 0.533·18-s − 0.522·19-s + 2.45·20-s − 1.82·22-s + 1.77·23-s + 0.516·24-s + 1.47·25-s + 0.648·26-s + 0.192·27-s − 0.937·29-s + 1.45·30-s − 0.260·31-s − 1.09·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.703881042$
$L(\frac12)$  $\approx$  $8.703881042$
$L(\frac{3}{2})$   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 \( 1 - 2.26T + 2T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 - 6.67T + 17T^{2} \)
19 \( 1 + 2.27T + 19T^{2} \)
23 \( 1 - 8.49T + 23T^{2} \)
29 \( 1 + 5.04T + 29T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 + 0.996T + 37T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 0.748T + 47T^{2} \)
53 \( 1 - 9.47T + 53T^{2} \)
59 \( 1 + 9.62T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 8.40T + 67T^{2} \)
71 \( 1 + 6.50T + 71T^{2} \)
73 \( 1 + 2.28T + 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 1.84T + 83T^{2} \)
89 \( 1 + 1.76T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−7.80922442245563473522704500352, −7.21068890764686208365703667710, −6.33368996377254908976077324722, −5.65305115684876233225935979130, −5.34523915984992476753841768098, −4.57506148387087042167797365429, −3.52029626934069609130672060099, −2.88913693893690821116748999003, −2.29122658309934533928450792333, −1.33580155952963031757416730989, 1.33580155952963031757416730989, 2.29122658309934533928450792333, 2.88913693893690821116748999003, 3.52029626934069609130672060099, 4.57506148387087042167797365429, 5.34523915984992476753841768098, 5.65305115684876233225935979130, 6.33368996377254908976077324722, 7.21068890764686208365703667710, 7.80922442245563473522704500352

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