Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s − 6·13-s + 14-s + 16-s + 6·19-s + 20-s − 22-s − 4·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 4·31-s − 32-s − 35-s − 12·37-s − 6·38-s − 40-s + 2·41-s + 6·43-s + 44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.169·35-s − 1.97·37-s − 0.973·38-s − 0.158·40-s + 0.312·41-s + 0.914·43-s + 0.150·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6930} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.48186317272435, −17.15167086893407, −16.19810580960906, −16.05109565866135, −15.21728102803109, −14.52332791418967, −14.04225817122046, −13.42954621302958, −12.51050505688434, −12.04157461997989, −11.64333303787345, −10.60886013838009, −10.07318494009915, −9.653346374595545, −9.114387769730193, −8.310795579107646, −7.554604077306944, −7.038550466151578, −6.384850016474816, −5.501111945140349, −4.945493161343597, −3.859605437351527, −2.908947214328968, −2.253283869415899, −1.215859982147239, 0, 1.215859982147239, 2.253283869415899, 2.908947214328968, 3.859605437351527, 4.945493161343597, 5.501111945140349, 6.384850016474816, 7.038550466151578, 7.554604077306944, 8.310795579107646, 9.114387769730193, 9.653346374595545, 10.07318494009915, 10.60886013838009, 11.64333303787345, 12.04157461997989, 12.51050505688434, 13.42954621302958, 14.04225817122046, 14.52332791418967, 15.21728102803109, 16.05109565866135, 16.19810580960906, 17.15167086893407, 17.48186317272435

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