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 − 2·19-s + 20-s + 22-s + 4·23-s + 25-s − 2·26-s − 28-s + 10·29-s − 4·31-s − 32-s − 35-s − 4·37-s + 2·38-s − 40-s − 6·41-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-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 + 1.85·29-s − 0.718·31-s − 0.176·32-s − 0.169·35-s − 0.657·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-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  =  0
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.419032292$
$L(\frac12)$  $\approx$  $1.419032292$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−17.34124813263006, −16.57128345441802, −16.18704532281831, −15.56784468093903, −14.95898514814864, −14.30335945091966, −13.55418891419180, −13.08458296070814, −12.38275741872716, −11.78332706333415, −10.93398884646228, −10.53255177642546, −9.907616923197090, −9.291822376556934, −8.516878191353050, −8.263783043118446, −7.062771723526893, −6.804973175667705, −5.941772249065958, −5.299579656891613, −4.367078794621005, −3.347115598187702, −2.645366520943412, −1.699617322886936, −0.6850898910309890, 0.6850898910309890, 1.699617322886936, 2.645366520943412, 3.347115598187702, 4.367078794621005, 5.299579656891613, 5.941772249065958, 6.804973175667705, 7.062771723526893, 8.263783043118446, 8.516878191353050, 9.291822376556934, 9.907616923197090, 10.53255177642546, 10.93398884646228, 11.78332706333415, 12.38275741872716, 13.08458296070814, 13.55418891419180, 14.30335945091966, 14.95898514814864, 15.56784468093903, 16.18704532281831, 16.57128345441802, 17.34124813263006

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