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 − 4·13-s − 14-s + 16-s − 4·19-s + 20-s − 22-s + 25-s + 4·26-s + 28-s + 6·29-s − 10·31-s − 32-s + 35-s + 2·37-s + 4·38-s − 40-s + 12·41-s − 4·43-s + 44-s − 6·47-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 − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.87·41-s − 0.609·43-s + 0.150·44-s − 0.875·47-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.60725357100113, −16.91901298096739, −16.53032878278654, −15.91362085464608, −14.94569926269260, −14.72497251682921, −14.14491342289589, −13.27059854037228, −12.61426948965253, −12.14372612877894, −11.31296263029785, −10.85867798298630, −10.08711594409582, −9.670421047383744, −8.892904780760043, −8.449131198258432, −7.537203256156622, −7.107308504958430, −6.282730152670679, −5.616847619805805, −4.777268931026782, −4.027321234774567, −2.849910051254840, −2.178034740146779, −1.302316315845907, 0, 1.302316315845907, 2.178034740146779, 2.849910051254840, 4.027321234774567, 4.777268931026782, 5.616847619805805, 6.282730152670679, 7.107308504958430, 7.537203256156622, 8.449131198258432, 8.892904780760043, 9.670421047383744, 10.08711594409582, 10.85867798298630, 11.31296263029785, 12.14372612877894, 12.61426948965253, 13.27059854037228, 14.14491342289589, 14.72497251682921, 14.94569926269260, 15.91362085464608, 16.53032878278654, 16.91901298096739, 17.60725357100113

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