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  − 0.442·2-s + 3-s − 1.80·4-s − 4.29·5-s − 0.442·6-s + 1.68·8-s + 9-s + 1.89·10-s − 2.21·11-s − 1.80·12-s − 3.11·13-s − 4.29·15-s + 2.86·16-s − 6.92·17-s − 0.442·18-s + 7.72·19-s + 7.74·20-s + 0.978·22-s + 0.0381·23-s + 1.68·24-s + 13.4·25-s + 1.37·26-s + 27-s + 2.66·29-s + 1.89·30-s + 4.69·31-s − 4.63·32-s + ⋯
L(s)  = 1  − 0.312·2-s + 0.577·3-s − 0.902·4-s − 1.91·5-s − 0.180·6-s + 0.594·8-s + 0.333·9-s + 0.600·10-s − 0.666·11-s − 0.520·12-s − 0.864·13-s − 1.10·15-s + 0.716·16-s − 1.67·17-s − 0.104·18-s + 1.77·19-s + 1.73·20-s + 0.208·22-s + 0.00795·23-s + 0.343·24-s + 2.68·25-s + 0.270·26-s + 0.192·27-s + 0.495·29-s + 0.346·30-s + 0.844·31-s − 0.818·32-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 \( 1 + 0.442T + 2T^{2} \)
5 \( 1 + 4.29T + 5T^{2} \)
11 \( 1 + 2.21T + 11T^{2} \)
13 \( 1 + 3.11T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 7.72T + 19T^{2} \)
23 \( 1 - 0.0381T + 23T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 - 4.69T + 31T^{2} \)
37 \( 1 + 0.0300T + 37T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 3.78T + 47T^{2} \)
53 \( 1 + 0.546T + 53T^{2} \)
59 \( 1 + 9.55T + 59T^{2} \)
61 \( 1 - 6.02T + 61T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + 2.42T + 73T^{2} \)
79 \( 1 - 2.16T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 8.69T + 89T^{2} \)
97 \( 1 - 0.216T + 97T^{2} \)
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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

−7.74327764794378990734580943653, −7.46596778731140360207509942607, −6.62763629893622827800536983124, −5.17207397414629900028202294908, −4.67552777460138067795529286899, −4.10273944612347107835549044043, −3.29987537166387006555194103699, −2.55421316633116235771750547873, −0.933126193624121192041351459681, 0, 0.933126193624121192041351459681, 2.55421316633116235771750547873, 3.29987537166387006555194103699, 4.10273944612347107835549044043, 4.67552777460138067795529286899, 5.17207397414629900028202294908, 6.62763629893622827800536983124, 7.46596778731140360207509942607, 7.74327764794378990734580943653

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