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 + 2·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 20-s + 22-s + 25-s + 2·26-s + 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s − 35-s + 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−17.29693088416659, −16.91945059114195, −16.18622005150373, −15.61021966937468, −15.04646363199922, −14.69221240808235, −13.90952039513388, −13.27044545493589, −12.88762414712262, −12.15776705579305, −11.40335374567947, −11.10016777094911, −10.53727951238383, −9.561548641579912, −8.795321666217083, −8.314372493206106, −7.483298042081545, −6.783414498650076, −6.250562928190339, −5.410234353768901, −4.612689386913835, −4.054407462288863, −3.376705543944651, −2.294634612043046, −1.552043400385183, 0, 1.552043400385183, 2.294634612043046, 3.376705543944651, 4.054407462288863, 4.612689386913835, 5.410234353768901, 6.250562928190339, 6.783414498650076, 7.483298042081545, 8.314372493206106, 8.795321666217083, 9.561548641579912, 10.53727951238383, 11.10016777094911, 11.40335374567947, 12.15776705579305, 12.88762414712262, 13.27044545493589, 13.90952039513388, 14.69221240808235, 15.04646363199922, 15.61021966937468, 16.18622005150373, 16.91945059114195, 17.29693088416659

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