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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.610481359$
$L(\frac12)$  $\approx$  $1.610481359$
$L(\frac{3}{2})$   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 \( 1 + 0.844T + 2T^{2} \)
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

−8.103394376177089630245563788362, −7.73621394462638058324097397525, −6.93444944544294505203708400880, −6.01002523195246073173783965294, −5.02680909967248695024244838322, −4.32140186719471406825139558285, −3.58249294919847640313519381896, −3.03140742384587357961413488321, −1.44663535910573927500183546849, −0.818253767334180792473143545459, 0.818253767334180792473143545459, 1.44663535910573927500183546849, 3.03140742384587357961413488321, 3.58249294919847640313519381896, 4.32140186719471406825139558285, 5.02680909967248695024244838322, 6.01002523195246073173783965294, 6.93444944544294505203708400880, 7.73621394462638058324097397525, 8.103394376177089630245563788362

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