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  − 2.72·2-s + 3-s + 5.43·4-s − 0.416·5-s − 2.72·6-s − 9.37·8-s + 9-s + 1.13·10-s − 0.0467·11-s + 5.43·12-s + 3.00·13-s − 0.416·15-s + 14.6·16-s − 5.85·17-s − 2.72·18-s + 6.05·19-s − 2.26·20-s + 0.127·22-s + 0.952·23-s − 9.37·24-s − 4.82·25-s − 8.19·26-s + 27-s + 2.00·29-s + 1.13·30-s − 1.80·31-s − 21.3·32-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577·3-s + 2.71·4-s − 0.186·5-s − 1.11·6-s − 3.31·8-s + 0.333·9-s + 0.358·10-s − 0.0141·11-s + 1.56·12-s + 0.833·13-s − 0.107·15-s + 3.67·16-s − 1.42·17-s − 0.642·18-s + 1.38·19-s − 0.506·20-s + 0.0272·22-s + 0.198·23-s − 1.91·24-s − 0.965·25-s − 1.60·26-s + 0.192·27-s + 0.372·29-s + 0.207·30-s − 0.323·31-s − 3.76·32-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 \( 1 + 2.72T + 2T^{2} \)
5 \( 1 + 0.416T + 5T^{2} \)
11 \( 1 + 0.0467T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
17 \( 1 + 5.85T + 17T^{2} \)
19 \( 1 - 6.05T + 19T^{2} \)
23 \( 1 - 0.952T + 23T^{2} \)
29 \( 1 - 2.00T + 29T^{2} \)
31 \( 1 + 1.80T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
43 \( 1 + 9.90T + 43T^{2} \)
47 \( 1 + 9.28T + 47T^{2} \)
53 \( 1 + 2.46T + 53T^{2} \)
59 \( 1 + 6.18T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 9.49T + 67T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 3.23T + 83T^{2} \)
89 \( 1 - 4.51T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−7.88102582879884913844871409579, −7.39998191692361144119633871508, −6.55845992983640799170693340356, −6.10179386379308892117350203743, −4.87273336334948691363289669123, −3.61022052012086679599592476532, −2.93970384382117588241181302253, −1.97416702812457642578341570100, −1.25465183899006123376368462299, 0, 1.25465183899006123376368462299, 1.97416702812457642578341570100, 2.93970384382117588241181302253, 3.61022052012086679599592476532, 4.87273336334948691363289669123, 6.10179386379308892117350203743, 6.55845992983640799170693340356, 7.39998191692361144119633871508, 7.88102582879884913844871409579

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