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  − 2.13·2-s + 3-s + 2.54·4-s + 2.91·5-s − 2.13·6-s − 1.16·8-s + 9-s − 6.21·10-s − 1.50·11-s + 2.54·12-s − 2.14·13-s + 2.91·15-s − 2.60·16-s + 2.46·17-s − 2.13·18-s − 7.23·19-s + 7.41·20-s + 3.19·22-s + 4.49·23-s − 1.16·24-s + 3.48·25-s + 4.57·26-s + 27-s − 9.11·29-s − 6.21·30-s − 7.76·31-s + 7.89·32-s + ⋯
L(s)  = 1  − 1.50·2-s + 0.577·3-s + 1.27·4-s + 1.30·5-s − 0.870·6-s − 0.411·8-s + 0.333·9-s − 1.96·10-s − 0.452·11-s + 0.734·12-s − 0.595·13-s + 0.752·15-s − 0.652·16-s + 0.598·17-s − 0.502·18-s − 1.65·19-s + 1.65·20-s + 0.682·22-s + 0.937·23-s − 0.237·24-s + 0.697·25-s + 0.897·26-s + 0.192·27-s − 1.69·29-s − 1.13·30-s − 1.39·31-s + 1.39·32-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 \( 1 + 2.13T + 2T^{2} \)
5 \( 1 - 2.91T + 5T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
13 \( 1 + 2.14T + 13T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 - 4.49T + 23T^{2} \)
29 \( 1 + 9.11T + 29T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 - 1.76T + 37T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 1.43T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 + 7.69T + 59T^{2} \)
61 \( 1 - 5.74T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 - 2.78T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 1.23T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 2.93T + 97T^{2} \)
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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

−7.75757232754953681322955071497, −7.35492498199508823720529010344, −6.54670909053701856732083101112, −5.75571541294411366151615453999, −4.98273262214845720474676891889, −3.91078134595025193758383140709, −2.67073674858921380411556269077, −2.10087651922802864112537081414, −1.41650765711262244401455228205, 0, 1.41650765711262244401455228205, 2.10087651922802864112537081414, 2.67073674858921380411556269077, 3.91078134595025193758383140709, 4.98273262214845720474676891889, 5.75571541294411366151615453999, 6.54670909053701856732083101112, 7.35492498199508823720529010344, 7.75757232754953681322955071497

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