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  + 2.56·2-s + 3-s + 4.56·4-s − 3.89·5-s + 2.56·6-s + 6.57·8-s + 9-s − 9.97·10-s + 3.81·11-s + 4.56·12-s + 6.39·13-s − 3.89·15-s + 7.71·16-s + 1.70·17-s + 2.56·18-s − 6.47·19-s − 17.7·20-s + 9.77·22-s + 3.51·23-s + 6.57·24-s + 10.1·25-s + 16.3·26-s + 27-s − 8.65·29-s − 9.97·30-s + 0.432·31-s + 6.62·32-s + ⋯
L(s)  = 1  + 1.81·2-s + 0.577·3-s + 2.28·4-s − 1.74·5-s + 1.04·6-s + 2.32·8-s + 0.333·9-s − 3.15·10-s + 1.14·11-s + 1.31·12-s + 1.77·13-s − 1.00·15-s + 1.92·16-s + 0.413·17-s + 0.603·18-s − 1.48·19-s − 3.97·20-s + 2.08·22-s + 0.732·23-s + 1.34·24-s + 2.03·25-s + 3.21·26-s + 0.192·27-s − 1.60·29-s − 1.82·30-s + 0.0776·31-s + 1.17·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.916748483$
$L(\frac12)$  $\approx$  $6.916748483$
$L(\frac{3}{2})$   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 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 + 3.89T + 5T^{2} \)
11 \( 1 - 3.81T + 11T^{2} \)
13 \( 1 - 6.39T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 - 0.432T + 31T^{2} \)
37 \( 1 - 7.97T + 37T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 - 3.72T + 47T^{2} \)
53 \( 1 - 14.2T + 53T^{2} \)
59 \( 1 - 3.38T + 59T^{2} \)
61 \( 1 + 4.40T + 61T^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 + 0.619T + 71T^{2} \)
73 \( 1 - 2.71T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 5.74T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 2.81T + 97T^{2} \)
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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

−7.892323254890860660470571028993, −7.17658758017563867400976248503, −6.54916262676089230135189189360, −5.96348134906135356904656993260, −4.88895683002484532255383741739, −4.09798693315493040175698904373, −3.74648315947162012933360433727, −3.42168894534205278742268489009, −2.25592734843751078150504839733, −1.09312919846010506766258195739, 1.09312919846010506766258195739, 2.25592734843751078150504839733, 3.42168894534205278742268489009, 3.74648315947162012933360433727, 4.09798693315493040175698904373, 4.88895683002484532255383741739, 5.96348134906135356904656993260, 6.54916262676089230135189189360, 7.17658758017563867400976248503, 7.892323254890860660470571028993

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