Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \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 + 4-s + 5-s + 8-s + 10-s + 11-s + 6·13-s + 16-s − 17-s − 4·19-s + 20-s + 22-s + 2·23-s + 25-s + 6·26-s + 10·29-s + 4·31-s + 32-s − 34-s + 10·37-s − 4·38-s + 40-s − 6·41-s + 8·43-s + 44-s + 2·46-s + 4·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s + 1.17·26-s + 1.85·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + 1.64·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s + 0.150·44-s + 0.294·46-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.890649699$
$L(\frac12)$  $\approx$  $4.890649699$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.69970374103367, −15.50079105734516, −14.61778209779973, −14.21298624796720, −13.67348995962975, −13.17356020666718, −12.72104677229128, −12.10584083189917, −11.35179524609002, −10.98162615795495, −10.43764686359519, −9.731803100975418, −9.066299080753578, −8.331738769046876, −8.062251274405187, −6.883690369870055, −6.471911433431489, −6.067047165603403, −5.351571515525988, −4.427428766181761, −4.163396501961486, −3.150334471419809, −2.621472694036141, −1.612805290847086, −0.9047093099379170, 0.9047093099379170, 1.612805290847086, 2.621472694036141, 3.150334471419809, 4.163396501961486, 4.427428766181761, 5.351571515525988, 6.067047165603403, 6.471911433431489, 6.883690369870055, 8.062251274405187, 8.331738769046876, 9.066299080753578, 9.731803100975418, 10.43764686359519, 10.98162615795495, 11.35179524609002, 12.10584083189917, 12.72104677229128, 13.17356020666718, 13.67348995962975, 14.21298624796720, 14.61778209779973, 15.50079105734516, 15.69970374103367

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