Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 $
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 − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 4·19-s − 20-s + 21-s + 24-s + 25-s − 26-s − 27-s − 28-s + 2·29-s − 30-s + 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(46410\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{46410} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 46410,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.169828879$
$L(\frac12)$  $\approx$  $1.169828879$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
good11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.81968787553489, −14.01634616571320, −13.55773273722573, −13.01272898454893, −12.31961212637042, −11.92957947246297, −11.54547121838828, −10.87279781199627, −10.53749758004785, −9.889909496166610, −9.405957015327981, −8.929778912209931, −8.132151977758343, −7.843924471751380, −7.125572869056453, −6.619791521406545, −6.167542296928002, −5.456988417870716, −4.866101600292656, −4.140827310322672, −3.465650394540033, −2.824210815982511, −2.028789109209066, −1.064370539255773, −0.5380642537883288, 0.5380642537883288, 1.064370539255773, 2.028789109209066, 2.824210815982511, 3.465650394540033, 4.140827310322672, 4.866101600292656, 5.456988417870716, 6.167542296928002, 6.619791521406545, 7.125572869056453, 7.843924471751380, 8.132151977758343, 8.929778912209931, 9.405957015327981, 9.889909496166610, 10.53749758004785, 10.87279781199627, 11.54547121838828, 11.92957947246297, 12.31961212637042, 13.01272898454893, 13.55773273722573, 14.01634616571320, 14.81968787553489

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