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  − 1.80·2-s + 3-s + 1.24·4-s − 2.17·5-s − 1.80·6-s + 1.36·8-s + 9-s + 3.92·10-s − 0.798·11-s + 1.24·12-s + 1.19·13-s − 2.17·15-s − 4.94·16-s − 5.17·17-s − 1.80·18-s − 4.47·19-s − 2.70·20-s + 1.43·22-s + 6.71·23-s + 1.36·24-s − 0.257·25-s − 2.14·26-s + 27-s + 4.22·29-s + 3.92·30-s + 0.168·31-s + 6.17·32-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.577·3-s + 0.621·4-s − 0.973·5-s − 0.735·6-s + 0.482·8-s + 0.333·9-s + 1.24·10-s − 0.240·11-s + 0.358·12-s + 0.330·13-s − 0.562·15-s − 1.23·16-s − 1.25·17-s − 0.424·18-s − 1.02·19-s − 0.605·20-s + 0.306·22-s + 1.40·23-s + 0.278·24-s − 0.0514·25-s − 0.421·26-s + 0.192·27-s + 0.785·29-s + 0.716·30-s + 0.0302·31-s + 1.09·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6422684099$
$L(\frac12)$  $\approx$  $0.6422684099$
$L(\frac{3}{2})$   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 \( 1 + 1.80T + 2T^{2} \)
5 \( 1 + 2.17T + 5T^{2} \)
11 \( 1 + 0.798T + 11T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 + 5.17T + 17T^{2} \)
19 \( 1 + 4.47T + 19T^{2} \)
23 \( 1 - 6.71T + 23T^{2} \)
29 \( 1 - 4.22T + 29T^{2} \)
31 \( 1 - 0.168T + 31T^{2} \)
37 \( 1 + 5.83T + 37T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 + 5.26T + 47T^{2} \)
53 \( 1 + 1.21T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 2.41T + 67T^{2} \)
71 \( 1 - 0.255T + 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 - 4.79T + 97T^{2} \)
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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

−8.256116602382816141084959467099, −7.61347184143685293885119733973, −6.95973915443972346631795282383, −6.39119489198581307786002759462, −4.95471045612696651662005633311, −4.39283535629618813888454289228, −3.58198144084876028495136880669, −2.57943130841245353521173678613, −1.67553326541656400356422824774, −0.49854555152394373073977689573, 0.49854555152394373073977689573, 1.67553326541656400356422824774, 2.57943130841245353521173678613, 3.58198144084876028495136880669, 4.39283535629618813888454289228, 4.95471045612696651662005633311, 6.39119489198581307786002759462, 6.95973915443972346631795282383, 7.61347184143685293885119733973, 8.256116602382816141084959467099

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