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  + 1.41·2-s − 3-s − 0.585·5-s − 1.41·6-s − 2.82·8-s + 9-s − 0.828·10-s − 0.414·11-s + 2.24·13-s + 0.585·15-s − 4.00·16-s − 2.41·17-s + 1.41·18-s + 2.58·19-s − 0.585·22-s + 1.41·23-s + 2.82·24-s − 4.65·25-s + 3.17·26-s − 27-s + 8.07·29-s + 0.828·30-s + 3·31-s + 0.414·33-s − 3.41·34-s + ⋯
L(s)  = 1  + 1.00·2-s − 0.577·3-s − 0.261·5-s − 0.577·6-s − 0.999·8-s + 0.333·9-s − 0.261·10-s − 0.124·11-s + 0.621·13-s + 0.151·15-s − 1.00·16-s − 0.585·17-s + 0.333·18-s + 0.593·19-s − 0.124·22-s + 0.294·23-s + 0.577·24-s − 0.931·25-s + 0.621·26-s − 0.192·27-s + 1.49·29-s + 0.151·30-s + 0.538·31-s + 0.0721·33-s − 0.585·34-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 \( 1 - 1.41T + 2T^{2} \)
5 \( 1 + 0.585T + 5T^{2} \)
11 \( 1 + 0.414T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 + 2.41T + 17T^{2} \)
19 \( 1 - 2.58T + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 7.48T + 37T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 6.65T + 61T^{2} \)
67 \( 1 + 6.48T + 67T^{2} \)
71 \( 1 + 4.07T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 3.65T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 0.343T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−7.68542676738369525551702232502, −6.62297992810949274203209241665, −6.23042952138944208166638268726, −5.50118210440411029203220875502, −4.70513986997764905516749695502, −4.29492636100223364127288075987, −3.38508879765128773443871102874, −2.65769814446075140339979762284, −1.26491025786676716303416602889, 0, 1.26491025786676716303416602889, 2.65769814446075140339979762284, 3.38508879765128773443871102874, 4.29492636100223364127288075987, 4.70513986997764905516749695502, 5.50118210440411029203220875502, 6.23042952138944208166638268726, 6.62297992810949274203209241665, 7.68542676738369525551702232502

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