Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.819·2-s + 3-s − 1.32·4-s + 0.332·5-s + 0.819·6-s − 2.72·8-s + 9-s + 0.272·10-s + 1.66·11-s − 1.32·12-s + 0.0164·13-s + 0.332·15-s + 0.418·16-s + 1.78·17-s + 0.819·18-s − 2.53·19-s − 0.440·20-s + 1.36·22-s − 2.87·23-s − 2.72·24-s − 4.88·25-s + 0.0134·26-s + 27-s + 0.605·29-s + 0.272·30-s − 7.60·31-s + 5.79·32-s + ⋯
L(s)  = 1  + 0.579·2-s + 0.577·3-s − 0.663·4-s + 0.148·5-s + 0.334·6-s − 0.964·8-s + 0.333·9-s + 0.0861·10-s + 0.501·11-s − 0.383·12-s + 0.00456·13-s + 0.0857·15-s + 0.104·16-s + 0.433·17-s + 0.193·18-s − 0.581·19-s − 0.0985·20-s + 0.290·22-s − 0.599·23-s − 0.556·24-s − 0.977·25-s + 0.00264·26-s + 0.192·27-s + 0.112·29-s + 0.0497·30-s − 1.36·31-s + 1.02·32-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 0.819T + 2T^{2} \)
5 \( 1 - 0.332T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 - 0.0164T + 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 2.53T + 19T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 - 0.605T + 29T^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 - 1.91T + 37T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 - 3.84T + 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 4.42T + 61T^{2} \)
67 \( 1 - 4.72T + 67T^{2} \)
71 \( 1 - 5.80T + 71T^{2} \)
73 \( 1 - 0.748T + 73T^{2} \)
79 \( 1 - 4.49T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 - 6.40T + 97T^{2} \)
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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

−7.962570478826541315728576221715, −6.90990908893516110130108206822, −6.22069169641845066998742506141, −5.46909613728436582338164418642, −4.77445254160006032728785749265, −3.84523721789232339561250690319, −3.56849825483040451970113021035, −2.45632415298573175529501137685, −1.48849318474692407283212431796, 0, 1.48849318474692407283212431796, 2.45632415298573175529501137685, 3.56849825483040451970113021035, 3.84523721789232339561250690319, 4.77445254160006032728785749265, 5.46909613728436582338164418642, 6.22069169641845066998742506141, 6.90990908893516110130108206822, 7.962570478826541315728576221715

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