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  − 1.21·2-s + 3-s − 0.534·4-s − 0.826·5-s − 1.21·6-s + 3.06·8-s + 9-s + 1.00·10-s − 0.396·11-s − 0.534·12-s − 3.17·13-s − 0.826·15-s − 2.64·16-s − 0.159·17-s − 1.21·18-s − 3.91·19-s + 0.441·20-s + 0.480·22-s + 7.13·23-s + 3.06·24-s − 4.31·25-s + 3.84·26-s + 27-s + 9.77·29-s + 1.00·30-s − 6.15·31-s − 2.93·32-s + ⋯
L(s)  = 1  − 0.856·2-s + 0.577·3-s − 0.267·4-s − 0.369·5-s − 0.494·6-s + 1.08·8-s + 0.333·9-s + 0.316·10-s − 0.119·11-s − 0.154·12-s − 0.879·13-s − 0.213·15-s − 0.661·16-s − 0.0385·17-s − 0.285·18-s − 0.898·19-s + 0.0988·20-s + 0.102·22-s + 1.48·23-s + 0.626·24-s − 0.863·25-s + 0.753·26-s + 0.192·27-s + 1.81·29-s + 0.182·30-s − 1.10·31-s − 0.518·32-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 \( 1 + 1.21T + 2T^{2} \)
5 \( 1 + 0.826T + 5T^{2} \)
11 \( 1 + 0.396T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 + 0.159T + 17T^{2} \)
19 \( 1 + 3.91T + 19T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 - 9.77T + 29T^{2} \)
31 \( 1 + 6.15T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
43 \( 1 + 5.32T + 43T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 + 5.54T + 53T^{2} \)
59 \( 1 + 1.57T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 1.25T + 71T^{2} \)
73 \( 1 + 8.34T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 + 9.77T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−7.943551807567387734579502459503, −7.22469376240437062479005549181, −6.68082106537282012700934527594, −5.48019750133830327454653025699, −4.63103799388543522153571827264, −4.16485209712632947527520029645, −3.08127075424410409927863243272, −2.24337426657071445423425871729, −1.15705411750442199800909476768, 0, 1.15705411750442199800909476768, 2.24337426657071445423425871729, 3.08127075424410409927863243272, 4.16485209712632947527520029645, 4.63103799388543522153571827264, 5.48019750133830327454653025699, 6.68082106537282012700934527594, 7.22469376240437062479005549181, 7.943551807567387734579502459503

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