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  + 0.712·2-s + 3-s − 1.49·4-s − 0.415·5-s + 0.712·6-s − 2.48·8-s + 9-s − 0.295·10-s + 2.75·11-s − 1.49·12-s − 4.20·13-s − 0.415·15-s + 1.21·16-s + 4.39·17-s + 0.712·18-s − 1.26·19-s + 0.619·20-s + 1.96·22-s + 4.53·23-s − 2.48·24-s − 4.82·25-s − 2.99·26-s + 27-s − 5.42·29-s − 0.295·30-s − 8.86·31-s + 5.84·32-s + ⋯
L(s)  = 1  + 0.503·2-s + 0.577·3-s − 0.746·4-s − 0.185·5-s + 0.290·6-s − 0.879·8-s + 0.333·9-s − 0.0935·10-s + 0.831·11-s − 0.430·12-s − 1.16·13-s − 0.107·15-s + 0.302·16-s + 1.06·17-s + 0.167·18-s − 0.290·19-s + 0.138·20-s + 0.419·22-s + 0.944·23-s − 0.508·24-s − 0.965·25-s − 0.587·26-s + 0.192·27-s − 1.00·29-s − 0.0540·30-s − 1.59·31-s + 1.03·32-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 \( 1 - 0.712T + 2T^{2} \)
5 \( 1 + 0.415T + 5T^{2} \)
11 \( 1 - 2.75T + 11T^{2} \)
13 \( 1 + 4.20T + 13T^{2} \)
17 \( 1 - 4.39T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 4.53T + 23T^{2} \)
29 \( 1 + 5.42T + 29T^{2} \)
31 \( 1 + 8.86T + 31T^{2} \)
37 \( 1 + 2.60T + 37T^{2} \)
43 \( 1 - 4.38T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + 1.43T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 - 2.83T + 71T^{2} \)
73 \( 1 + 3.91T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 4.04T + 83T^{2} \)
89 \( 1 - 1.02T + 89T^{2} \)
97 \( 1 - 7.88T + 97T^{2} \)
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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

−7.45209810765321284468848273935, −7.38896284746862864238025183657, −6.11432026263251391960239813546, −5.47331087600216845258013620758, −4.74262338638421627072978755630, −3.93006114114935872748011592794, −3.47550609597114637571955442911, −2.52945263807440330663330456446, −1.39355978312033456804145312181, 0, 1.39355978312033456804145312181, 2.52945263807440330663330456446, 3.47550609597114637571955442911, 3.93006114114935872748011592794, 4.74262338638421627072978755630, 5.47331087600216845258013620758, 6.11432026263251391960239813546, 7.38896284746862864238025183657, 7.45209810765321284468848273935

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