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  − 1.44·2-s + 3-s + 0.0739·4-s + 3.20·5-s − 1.44·6-s + 2.77·8-s + 9-s − 4.62·10-s − 3.71·11-s + 0.0739·12-s + 1.13·13-s + 3.20·15-s − 4.14·16-s − 5.67·17-s − 1.44·18-s + 3.14·19-s + 0.237·20-s + 5.35·22-s − 4.86·23-s + 2.77·24-s + 5.30·25-s − 1.63·26-s + 27-s + 2.25·29-s − 4.62·30-s − 1.73·31-s + 0.418·32-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.577·3-s + 0.0369·4-s + 1.43·5-s − 0.587·6-s + 0.980·8-s + 0.333·9-s − 1.46·10-s − 1.12·11-s + 0.0213·12-s + 0.314·13-s + 0.828·15-s − 1.03·16-s − 1.37·17-s − 0.339·18-s + 0.722·19-s + 0.0530·20-s + 1.14·22-s − 1.01·23-s + 0.566·24-s + 1.06·25-s − 0.319·26-s + 0.192·27-s + 0.417·29-s − 0.843·30-s − 0.312·31-s + 0.0738·32-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 \( 1 + 1.44T + 2T^{2} \)
5 \( 1 - 3.20T + 5T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 - 1.13T + 13T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 - 2.25T + 29T^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 - 3.39T + 53T^{2} \)
59 \( 1 + 3.87T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 + 4.36T + 67T^{2} \)
71 \( 1 - 7.54T + 71T^{2} \)
73 \( 1 + 9.11T + 73T^{2} \)
79 \( 1 - 0.0474T + 79T^{2} \)
83 \( 1 + 8.03T + 83T^{2} \)
89 \( 1 + 9.30T + 89T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−8.009287478652370093524343346196, −7.14943915066535728956695781352, −6.50434282229821506530047371657, −5.55786930779715392615379245563, −4.95431092873262552259555531106, −4.04810433834267663750255868637, −2.85377991428892980646437023209, −2.08788756316044523366551353049, −1.45308374355944886051194253633, 0, 1.45308374355944886051194253633, 2.08788756316044523366551353049, 2.85377991428892980646437023209, 4.04810433834267663750255868637, 4.95431092873262552259555531106, 5.55786930779715392615379245563, 6.50434282229821506530047371657, 7.14943915066535728956695781352, 8.009287478652370093524343346196

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