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  + 2.32·2-s − 3-s + 3.41·4-s − 0.0978·5-s − 2.32·6-s + 3.28·8-s + 9-s − 0.227·10-s − 0.582·11-s − 3.41·12-s − 3.95·13-s + 0.0978·15-s + 0.810·16-s + 3.98·17-s + 2.32·18-s + 4.96·19-s − 0.333·20-s − 1.35·22-s + 7.14·23-s − 3.28·24-s − 4.99·25-s − 9.19·26-s − 27-s + 2.15·29-s + 0.227·30-s + 3.43·31-s − 4.67·32-s + ⋯
L(s)  = 1  + 1.64·2-s − 0.577·3-s + 1.70·4-s − 0.0437·5-s − 0.949·6-s + 1.15·8-s + 0.333·9-s − 0.0719·10-s − 0.175·11-s − 0.984·12-s − 1.09·13-s + 0.0252·15-s + 0.202·16-s + 0.967·17-s + 0.548·18-s + 1.13·19-s − 0.0745·20-s − 0.288·22-s + 1.48·23-s − 0.669·24-s − 0.998·25-s − 1.80·26-s − 0.192·27-s + 0.400·29-s + 0.0415·30-s + 0.616·31-s − 0.826·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.382316562$
$L(\frac12)$  $\approx$  $4.382316562$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 2.32T + 2T^{2} \)
5 \( 1 + 0.0978T + 5T^{2} \)
11 \( 1 + 0.582T + 11T^{2} \)
13 \( 1 + 3.95T + 13T^{2} \)
17 \( 1 - 3.98T + 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
23 \( 1 - 7.14T + 23T^{2} \)
29 \( 1 - 2.15T + 29T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 + 2.57T + 37T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 - 2.71T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 2.48T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 + 0.246T + 71T^{2} \)
73 \( 1 - 3.11T + 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 - 6.99T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−7.54857084691621810899586228526, −7.23964554046736289896572222242, −6.45630747381152320644215337865, −5.56409838411329778783194377855, −5.26866708142346318073099722426, −4.61799137593505958572485163976, −3.75244070943888777985450118221, −3.02208945120627413066884011123, −2.23677562015477554710393427098, −0.888067159259169045486882482614, 0.888067159259169045486882482614, 2.23677562015477554710393427098, 3.02208945120627413066884011123, 3.75244070943888777985450118221, 4.61799137593505958572485163976, 5.26866708142346318073099722426, 5.56409838411329778783194377855, 6.45630747381152320644215337865, 7.23964554046736289896572222242, 7.54857084691621810899586228526

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