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  + 0.368·2-s − 3-s − 1.86·4-s + 0.572·5-s − 0.368·6-s − 1.42·8-s + 9-s + 0.210·10-s + 5.73·11-s + 1.86·12-s + 3.53·13-s − 0.572·15-s + 3.20·16-s + 1.83·17-s + 0.368·18-s + 5.95·19-s − 1.06·20-s + 2.11·22-s + 3.12·23-s + 1.42·24-s − 4.67·25-s + 1.30·26-s − 27-s − 0.0143·29-s − 0.210·30-s − 6.00·31-s + 4.02·32-s + ⋯
L(s)  = 1  + 0.260·2-s − 0.577·3-s − 0.932·4-s + 0.255·5-s − 0.150·6-s − 0.502·8-s + 0.333·9-s + 0.0666·10-s + 1.72·11-s + 0.538·12-s + 0.981·13-s − 0.147·15-s + 0.801·16-s + 0.444·17-s + 0.0867·18-s + 1.36·19-s − 0.238·20-s + 0.449·22-s + 0.650·23-s + 0.290·24-s − 0.934·25-s + 0.255·26-s − 0.192·27-s − 0.00266·29-s − 0.0384·30-s − 1.07·31-s + 0.711·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.991078079$
$L(\frac12)$  $\approx$  $1.991078079$
$L(\frac{3}{2})$   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 \( 1 - 0.368T + 2T^{2} \)
5 \( 1 - 0.572T + 5T^{2} \)
11 \( 1 - 5.73T + 11T^{2} \)
13 \( 1 - 3.53T + 13T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 0.0143T + 29T^{2} \)
31 \( 1 + 6.00T + 31T^{2} \)
37 \( 1 - 7.68T + 37T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 - 6.07T + 47T^{2} \)
53 \( 1 + 0.555T + 53T^{2} \)
59 \( 1 - 7.46T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 3.12T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 8.02T + 73T^{2} \)
79 \( 1 - 5.75T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 2.84T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−8.078020515976314451758349937346, −7.30764128570297849543375173064, −6.47588843851640048252024731989, −5.74164432450263859735565621839, −5.43012524144796340995691794661, −4.26053736657210684608799197105, −3.91885798034331349300734744054, −3.06462010507970179051091964534, −1.47536978668526918133237860114, −0.846234258579901223435554344069, 0.846234258579901223435554344069, 1.47536978668526918133237860114, 3.06462010507970179051091964534, 3.91885798034331349300734744054, 4.26053736657210684608799197105, 5.43012524144796340995691794661, 5.74164432450263859735565621839, 6.47588843851640048252024731989, 7.30764128570297849543375173064, 8.078020515976314451758349937346

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