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  + 1.33·2-s − 3-s − 0.230·4-s + 3.44·5-s − 1.33·6-s − 2.96·8-s + 9-s + 4.58·10-s + 2.42·11-s + 0.230·12-s + 4.50·13-s − 3.44·15-s − 3.48·16-s + 4.56·17-s + 1.33·18-s + 1.29·19-s − 0.794·20-s + 3.23·22-s − 5.25·23-s + 2.96·24-s + 6.85·25-s + 5.99·26-s − 27-s + 6.83·29-s − 4.58·30-s − 2.39·31-s + 1.29·32-s + ⋯
L(s)  = 1  + 0.940·2-s − 0.577·3-s − 0.115·4-s + 1.54·5-s − 0.543·6-s − 1.04·8-s + 0.333·9-s + 1.44·10-s + 0.732·11-s + 0.0666·12-s + 1.25·13-s − 0.889·15-s − 0.871·16-s + 1.10·17-s + 0.313·18-s + 0.296·19-s − 0.177·20-s + 0.689·22-s − 1.09·23-s + 0.605·24-s + 1.37·25-s + 1.17·26-s − 0.192·27-s + 1.26·29-s − 0.836·30-s − 0.429·31-s + 0.229·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.648251289$
$L(\frac12)$  $\approx$  $3.648251289$
$L(\frac{3}{2})$   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 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 - 3.44T + 5T^{2} \)
11 \( 1 - 2.42T + 11T^{2} \)
13 \( 1 - 4.50T + 13T^{2} \)
17 \( 1 - 4.56T + 17T^{2} \)
19 \( 1 - 1.29T + 19T^{2} \)
23 \( 1 + 5.25T + 23T^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 + 2.39T + 31T^{2} \)
37 \( 1 + 0.323T + 37T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 - 1.20T + 59T^{2} \)
61 \( 1 - 4.84T + 61T^{2} \)
67 \( 1 - 3.77T + 67T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + 8.20T + 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 - 2.30T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.185794739730205375556388729406, −6.95004542308698606802750067107, −6.22380088603707924500086586170, −5.89943244256135343338258461498, −5.38159907663681709935526807578, −4.55031632829333834280170182096, −3.75650230744755510668315228031, −2.99290359951484195061227584755, −1.82404787870426290541987525823, −0.965807436635123582488555578292, 0.965807436635123582488555578292, 1.82404787870426290541987525823, 2.99290359951484195061227584755, 3.75650230744755510668315228031, 4.55031632829333834280170182096, 5.38159907663681709935526807578, 5.89943244256135343338258461498, 6.22380088603707924500086586170, 6.95004542308698606802750067107, 8.185794739730205375556388729406

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