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  + 1.10·2-s + 3-s − 0.779·4-s − 3.67·5-s + 1.10·6-s − 3.07·8-s + 9-s − 4.06·10-s + 1.52·11-s − 0.779·12-s + 3.67·13-s − 3.67·15-s − 1.83·16-s − 4.16·17-s + 1.10·18-s + 5.23·19-s + 2.86·20-s + 1.68·22-s + 1.53·23-s − 3.07·24-s + 8.52·25-s + 4.06·26-s + 27-s − 8.04·29-s − 4.06·30-s + 8.26·31-s + 4.11·32-s + ⋯
L(s)  = 1  + 0.781·2-s + 0.577·3-s − 0.389·4-s − 1.64·5-s + 0.450·6-s − 1.08·8-s + 0.333·9-s − 1.28·10-s + 0.458·11-s − 0.225·12-s + 1.01·13-s − 0.949·15-s − 0.457·16-s − 1.00·17-s + 0.260·18-s + 1.19·19-s + 0.641·20-s + 0.358·22-s + 0.320·23-s − 0.626·24-s + 1.70·25-s + 0.796·26-s + 0.192·27-s − 1.49·29-s − 0.741·30-s + 1.48·31-s + 0.727·32-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 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 + 3.67T + 5T^{2} \)
11 \( 1 - 1.52T + 11T^{2} \)
13 \( 1 - 3.67T + 13T^{2} \)
17 \( 1 + 4.16T + 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 8.04T + 29T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 + 9.47T + 37T^{2} \)
43 \( 1 - 5.19T + 43T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + 9.52T + 53T^{2} \)
59 \( 1 - 1.95T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 7.95T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 5.81T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 - 4.22T + 97T^{2} \)
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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

−7.85176514876347272344535598838, −7.00910357498959896074922837253, −6.35683189074657407626739566943, −5.33385086965676416970321812213, −4.57471148431419215842539944175, −3.94190207569860373134832332788, −3.51449367178187850169531799079, −2.81760178004310197483045991872, −1.26818324727627886633230621058, 0, 1.26818324727627886633230621058, 2.81760178004310197483045991872, 3.51449367178187850169531799079, 3.94190207569860373134832332788, 4.57471148431419215842539944175, 5.33385086965676416970321812213, 6.35683189074657407626739566943, 7.00910357498959896074922837253, 7.85176514876347272344535598838

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