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  + 0.766·2-s + 3-s − 1.41·4-s − 1.12·5-s + 0.766·6-s − 2.61·8-s + 9-s − 0.864·10-s + 0.505·11-s − 1.41·12-s + 6.75·13-s − 1.12·15-s + 0.821·16-s − 3.67·17-s + 0.766·18-s − 5.76·19-s + 1.59·20-s + 0.387·22-s − 4.22·23-s − 2.61·24-s − 3.72·25-s + 5.17·26-s + 27-s + 6.10·29-s − 0.864·30-s + 6.07·31-s + 5.85·32-s + ⋯
L(s)  = 1  + 0.541·2-s + 0.577·3-s − 0.706·4-s − 0.504·5-s + 0.312·6-s − 0.924·8-s + 0.333·9-s − 0.273·10-s + 0.152·11-s − 0.407·12-s + 1.87·13-s − 0.291·15-s + 0.205·16-s − 0.890·17-s + 0.180·18-s − 1.32·19-s + 0.356·20-s + 0.0826·22-s − 0.881·23-s − 0.533·24-s − 0.745·25-s + 1.01·26-s + 0.192·27-s + 1.13·29-s − 0.157·30-s + 1.09·31-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 \( 1 - 0.766T + 2T^{2} \)
5 \( 1 + 1.12T + 5T^{2} \)
11 \( 1 - 0.505T + 11T^{2} \)
13 \( 1 - 6.75T + 13T^{2} \)
17 \( 1 + 3.67T + 17T^{2} \)
19 \( 1 + 5.76T + 19T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 - 6.10T + 29T^{2} \)
31 \( 1 - 6.07T + 31T^{2} \)
37 \( 1 + 3.42T + 37T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + 7.61T + 47T^{2} \)
53 \( 1 - 3.36T + 53T^{2} \)
59 \( 1 - 8.83T + 59T^{2} \)
61 \( 1 - 9.19T + 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + 7.27T + 71T^{2} \)
73 \( 1 + 3.16T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.145341961475725990123672358203, −6.77542725038049031766844314982, −6.34884524987490329547297693991, −5.55459447305317850225675664453, −4.45310000268783959075647064336, −4.09402179684484561070040852954, −3.50489531105935926443402125501, −2.54155001581693189373179491158, −1.37043932474548682311062930715, 0, 1.37043932474548682311062930715, 2.54155001581693189373179491158, 3.50489531105935926443402125501, 4.09402179684484561070040852954, 4.45310000268783959075647064336, 5.55459447305317850225675664453, 6.34884524987490329547297693991, 6.77542725038049031766844314982, 8.145341961475725990123672358203

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