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 2

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 − 4·11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 20-s + 21-s + 4·22-s − 8·23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 6·29-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 − 1.20·11-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 − 0.223·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + ⋯

Functional equation

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

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  =  2
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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−15.35391501079980, −14.79667750350834, −14.07949364931684, −13.53694756368323, −12.77855471229881, −12.62797256520156, −11.76099037284620, −11.65546709909382, −10.79078214814218, −10.52300219634050, −9.860087311676097, −9.611746947475094, −8.776177116304345, −8.183321266094134, −7.665504645291156, −7.389888944615036, −6.525220293231291, −6.148807104748826, −5.347852299508481, −5.040626325355385, −4.028154834372023, −3.584328889997814, −2.672056680512992, −2.111963391127715, −1.218729943695046, 0, 0, 1.218729943695046, 2.111963391127715, 2.672056680512992, 3.584328889997814, 4.028154834372023, 5.040626325355385, 5.347852299508481, 6.148807104748826, 6.525220293231291, 7.389888944615036, 7.665504645291156, 8.183321266094134, 8.776177116304345, 9.611746947475094, 9.860087311676097, 10.52300219634050, 10.79078214814218, 11.65546709909382, 11.76099037284620, 12.62797256520156, 12.77855471229881, 13.53694756368323, 14.07949364931684, 14.79667750350834, 15.35391501079980

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