Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0650·2-s − 3-s − 1.99·4-s + 2.65·5-s + 0.0650·6-s + 0.259·8-s + 9-s − 0.172·10-s − 5.47·11-s + 1.99·12-s + 3.40·13-s − 2.65·15-s + 3.97·16-s + 6.28·17-s − 0.0650·18-s + 4.25·19-s − 5.29·20-s + 0.355·22-s − 1.06·23-s − 0.259·24-s + 2.04·25-s − 0.221·26-s − 27-s − 6.10·29-s + 0.172·30-s + 10.4·31-s − 0.778·32-s + ⋯
L(s)  = 1  − 0.0459·2-s − 0.577·3-s − 0.997·4-s + 1.18·5-s + 0.0265·6-s + 0.0918·8-s + 0.333·9-s − 0.0545·10-s − 1.64·11-s + 0.576·12-s + 0.943·13-s − 0.685·15-s + 0.993·16-s + 1.52·17-s − 0.0153·18-s + 0.976·19-s − 1.18·20-s + 0.0758·22-s − 0.222·23-s − 0.0530·24-s + 0.408·25-s − 0.0434·26-s − 0.192·27-s − 1.13·29-s + 0.0315·30-s + 1.88·31-s − 0.137·32-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  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.515699794$
$L(\frac12)$  $\approx$  $1.515699794$
$L(\frac{3}{2})$   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 \( 1 + 0.0650T + 2T^{2} \)
5 \( 1 - 2.65T + 5T^{2} \)
11 \( 1 + 5.47T + 11T^{2} \)
13 \( 1 - 3.40T + 13T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 + 6.10T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 3.83T + 37T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 8.14T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 6.51T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 9.32T + 79T^{2} \)
83 \( 1 - 5.03T + 83T^{2} \)
89 \( 1 + 8.62T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−7.940627445652079048651456077243, −7.65313700578150562165382192110, −6.34770705519093733768125529830, −5.76625025856994407798225759713, −5.30617977848672284578608859525, −4.76713530029188152344050149749, −3.61601062222478815607284944936, −2.85667607893493603101985021937, −1.61547584015165843537924531488, −0.71431518586463662859576292317, 0.71431518586463662859576292317, 1.61547584015165843537924531488, 2.85667607893493603101985021937, 3.61601062222478815607284944936, 4.76713530029188152344050149749, 5.30617977848672284578608859525, 5.76625025856994407798225759713, 6.34770705519093733768125529830, 7.65313700578150562165382192110, 7.940627445652079048651456077243

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