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  − 1.32·2-s − 3-s − 0.246·4-s − 4.03·5-s + 1.32·6-s + 2.97·8-s + 9-s + 5.34·10-s + 4.32·11-s + 0.246·12-s − 1.67·13-s + 4.03·15-s − 3.44·16-s + 1.53·17-s − 1.32·18-s − 4.63·19-s + 0.995·20-s − 5.72·22-s + 1.68·23-s − 2.97·24-s + 11.2·25-s + 2.22·26-s − 27-s + 6.19·29-s − 5.34·30-s + 7.82·31-s − 1.38·32-s + ⋯
L(s)  = 1  − 0.936·2-s − 0.577·3-s − 0.123·4-s − 1.80·5-s + 0.540·6-s + 1.05·8-s + 0.333·9-s + 1.68·10-s + 1.30·11-s + 0.0712·12-s − 0.465·13-s + 1.04·15-s − 0.861·16-s + 0.372·17-s − 0.312·18-s − 1.06·19-s + 0.222·20-s − 1.21·22-s + 0.350·23-s − 0.607·24-s + 2.25·25-s + 0.435·26-s − 0.192·27-s + 1.14·29-s − 0.975·30-s + 1.40·31-s − 0.245·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5350200432$
$L(\frac12)$  $\approx$  $0.5350200432$
$L(\frac{3}{2})$   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 \( 1 + 1.32T + 2T^{2} \)
5 \( 1 + 4.03T + 5T^{2} \)
11 \( 1 - 4.32T + 11T^{2} \)
13 \( 1 + 1.67T + 13T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 + 4.63T + 19T^{2} \)
23 \( 1 - 1.68T + 23T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 - 7.82T + 31T^{2} \)
37 \( 1 - 5.18T + 37T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 - 7.25T + 47T^{2} \)
53 \( 1 + 1.52T + 53T^{2} \)
59 \( 1 + 6.75T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 - 3.40T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 - 5.99T + 73T^{2} \)
79 \( 1 - 5.84T + 79T^{2} \)
83 \( 1 - 0.0883T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 2.82T + 97T^{2} \)
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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

−8.099779197236862024013229311631, −7.53789142180245225770150504849, −6.86719726196017621867633600808, −6.24380099413500542959669291409, −4.93591234270137903775168290750, −4.29722735195252997189487270019, −3.99533828239259021437307215552, −2.78103730441282172909348071082, −1.23704346066914761973163286684, −0.54489654852909615213263963486, 0.54489654852909615213263963486, 1.23704346066914761973163286684, 2.78103730441282172909348071082, 3.99533828239259021437307215552, 4.29722735195252997189487270019, 4.93591234270137903775168290750, 6.24380099413500542959669291409, 6.86719726196017621867633600808, 7.53789142180245225770150504849, 8.099779197236862024013229311631

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