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 1

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 + 6·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s − 6·22-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-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 + 1.80·11-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 + 0.229·19-s − 0.223·20-s − 0.218·21-s − 1.27·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-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  =  1
Selberg data  =  $(2,\ 46410,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.96425606770224, −14.46274158838932, −13.99117701171846, −13.25924287088063, −12.57203327625354, −12.04073119440655, −11.69919320370707, −11.35788239255348, −10.75881850921153, −10.19191808200709, −9.442084463076783, −9.341212425833651, −8.488879653547350, −8.073802411337411, −7.387046238185175, −6.864748084764662, −6.508816554447978, −5.715893560003967, −5.225308184364215, −4.439634760128378, −3.691237404201563, −3.467618464040953, −2.122495641681804, −1.664848513324087, −0.8936312244663594, 0, 0.8936312244663594, 1.664848513324087, 2.122495641681804, 3.467618464040953, 3.691237404201563, 4.439634760128378, 5.225308184364215, 5.715893560003967, 6.508816554447978, 6.864748084764662, 7.387046238185175, 8.073802411337411, 8.488879653547350, 9.341212425833651, 9.442084463076783, 10.19191808200709, 10.75881850921153, 11.35788239255348, 11.69919320370707, 12.04073119440655, 12.57203327625354, 13.25924287088063, 13.99117701171846, 14.46274158838932, 14.96425606770224

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