Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s − 7-s + 9-s − 2·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s + 2·19-s − 2·20-s − 21-s + 23-s + 25-s + 27-s + 2·28-s − 8·29-s − 31-s − 2·33-s − 35-s − 2·36-s + 3·37-s − 5·39-s + 7·41-s + 4·44-s + 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.179·31-s − 0.348·33-s − 0.169·35-s − 1/3·36-s + 0.493·37-s − 0.800·39-s + 1.09·41-s + 0.603·44-s + 0.149·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30345} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 30345,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.36239904861023, −14.64651543845644, −14.26063226497850, −13.97335029610309, −13.11352995893142, −12.84393451856283, −12.60610212465596, −11.71820341088437, −11.02735296217078, −10.35341771824640, −9.777855245333608, −9.406485147103156, −9.187019561983146, −8.207023713355286, −7.869006039480311, −7.269107010952537, −6.625989666936906, −5.696041124617278, −5.274782367995728, −4.732565153582354, −3.973336956429519, −3.369615262441099, −2.611096824962537, −2.041695542874978, −0.9252390693074910, 0, 0.9252390693074910, 2.041695542874978, 2.611096824962537, 3.369615262441099, 3.973336956429519, 4.732565153582354, 5.274782367995728, 5.696041124617278, 6.625989666936906, 7.269107010952537, 7.869006039480311, 8.207023713355286, 9.187019561983146, 9.406485147103156, 9.777855245333608, 10.35341771824640, 11.02735296217078, 11.71820341088437, 12.60610212465596, 12.84393451856283, 13.11352995893142, 13.97335029610309, 14.26063226497850, 14.64651543845644, 15.36239904861023

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