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 0

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 − 2·17-s + 4·19-s + 20-s − 22-s + 4·23-s + 25-s − 2·26-s − 28-s + 6·29-s − 32-s + 2·34-s − 35-s + 2·37-s − 4·38-s − 40-s − 6·41-s + 12·43-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 − 0.485·17-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.82·43-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  =  0
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.636177237$
$L(\frac12)$  $\approx$  $1.636177237$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−17.10736794933669, −16.81609009727152, −16.03811678727238, −15.59375454244738, −15.04389787266324, −14.03975328305960, −13.85818149152602, −12.92773482217876, −12.48412617025114, −11.60936232388057, −11.17142559960118, −10.41660357199356, −9.908261552962809, −9.155778960343793, −8.843879932863189, −7.999269703901244, −7.270064039136368, −6.586646178706958, −6.077154111900921, −5.239904256823969, −4.376488489100932, −3.328324091579576, −2.699492473662413, −1.619000253471359, −0.7612365854525905, 0.7612365854525905, 1.619000253471359, 2.699492473662413, 3.328324091579576, 4.376488489100932, 5.239904256823969, 6.077154111900921, 6.586646178706958, 7.270064039136368, 7.999269703901244, 8.843879932863189, 9.155778960343793, 9.908261552962809, 10.41660357199356, 11.17142559960118, 11.60936232388057, 12.48412617025114, 12.92773482217876, 13.85818149152602, 14.03975328305960, 15.04389787266324, 15.59375454244738, 16.03811678727238, 16.81609009727152, 17.10736794933669

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