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  + 2.11·2-s + 3-s + 2.47·4-s − 0.285·5-s + 2.11·6-s + 0.998·8-s + 9-s − 0.602·10-s − 3.38·11-s + 2.47·12-s + 0.720·13-s − 0.285·15-s − 2.83·16-s − 4.26·17-s + 2.11·18-s − 6.19·19-s − 0.704·20-s − 7.16·22-s − 3.13·23-s + 0.998·24-s − 4.91·25-s + 1.52·26-s + 27-s + 0.696·29-s − 0.602·30-s + 9.88·31-s − 7.98·32-s + ⋯
L(s)  = 1  + 1.49·2-s + 0.577·3-s + 1.23·4-s − 0.127·5-s + 0.863·6-s + 0.352·8-s + 0.333·9-s − 0.190·10-s − 1.02·11-s + 0.713·12-s + 0.199·13-s − 0.0735·15-s − 0.708·16-s − 1.03·17-s + 0.498·18-s − 1.42·19-s − 0.157·20-s − 1.52·22-s − 0.653·23-s + 0.203·24-s − 0.983·25-s + 0.298·26-s + 0.192·27-s + 0.129·29-s − 0.110·30-s + 1.77·31-s − 1.41·32-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 \( 1 - 2.11T + 2T^{2} \)
5 \( 1 + 0.285T + 5T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 - 0.720T + 13T^{2} \)
17 \( 1 + 4.26T + 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 - 0.696T + 29T^{2} \)
31 \( 1 - 9.88T + 31T^{2} \)
37 \( 1 - 0.244T + 37T^{2} \)
43 \( 1 + 2.01T + 43T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 - 9.03T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 2.17T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 - 5.06T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−7.74485495681790832934613945823, −6.60738665194966190141965506545, −6.35391750217875750645821807337, −5.42178960227201737978356507518, −4.57551185412073986828501776652, −4.22437845943882103421306745642, −3.32579397403601233495397620014, −2.54586363213468981231354880576, −1.94928147768142589503945219865, 0, 1.94928147768142589503945219865, 2.54586363213468981231354880576, 3.32579397403601233495397620014, 4.22437845943882103421306745642, 4.57551185412073986828501776652, 5.42178960227201737978356507518, 6.35391750217875750645821807337, 6.60738665194966190141965506545, 7.74485495681790832934613945823

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