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  − 0.844·2-s − 3-s − 1.28·4-s + 1.91·5-s + 0.844·6-s + 2.77·8-s + 9-s − 1.61·10-s + 2.68·11-s + 1.28·12-s − 1.87·13-s − 1.91·15-s + 0.231·16-s − 6.09·17-s − 0.844·18-s − 7.23·19-s − 2.46·20-s − 2.26·22-s − 3.42·23-s − 2.77·24-s − 1.34·25-s + 1.58·26-s − 27-s + 10.5·29-s + 1.61·30-s − 7.41·31-s − 5.74·32-s + ⋯
L(s)  = 1  − 0.596·2-s − 0.577·3-s − 0.643·4-s + 0.855·5-s + 0.344·6-s + 0.981·8-s + 0.333·9-s − 0.510·10-s + 0.808·11-s + 0.371·12-s − 0.521·13-s − 0.493·15-s + 0.0579·16-s − 1.47·17-s − 0.198·18-s − 1.66·19-s − 0.550·20-s − 0.482·22-s − 0.714·23-s − 0.566·24-s − 0.268·25-s + 0.311·26-s − 0.192·27-s + 1.95·29-s + 0.294·30-s − 1.33·31-s − 1.01·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8017389487$
$L(\frac12)$  $\approx$  $0.8017389487$
$L(\frac{3}{2})$   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 \( 1 + 0.844T + 2T^{2} \)
5 \( 1 - 1.91T + 5T^{2} \)
11 \( 1 - 2.68T + 11T^{2} \)
13 \( 1 + 1.87T + 13T^{2} \)
17 \( 1 + 6.09T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 - 1.51T + 37T^{2} \)
43 \( 1 + 4.03T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 1.76T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 + 4.62T + 71T^{2} \)
73 \( 1 + 4.04T + 73T^{2} \)
79 \( 1 - 1.99T + 79T^{2} \)
83 \( 1 - 7.18T + 83T^{2} \)
89 \( 1 + 4.71T + 89T^{2} \)
97 \( 1 - 4.66T + 97T^{2} \)
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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

−8.367675022534537220705240932335, −7.27778858211712628665322570985, −6.64700166140725552282599855900, −6.03027272100334651283305159387, −5.24910387821779513181614456156, −4.32685301629724765180927102096, −4.06587811593515271762121463329, −2.39328218944516950971855378507, −1.74531858948615903153832261885, −0.53401906489958562517152301149, 0.53401906489958562517152301149, 1.74531858948615903153832261885, 2.39328218944516950971855378507, 4.06587811593515271762121463329, 4.32685301629724765180927102096, 5.24910387821779513181614456156, 6.03027272100334651283305159387, 6.64700166140725552282599855900, 7.27778858211712628665322570985, 8.367675022534537220705240932335

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