Properties

Degree 2
Conductor $ 3^{2} \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  − 2·4-s − 3·5-s − 4·7-s + 3·11-s − 13-s + 4·16-s + 17-s − 19-s + 6·20-s − 9·23-s + 4·25-s + 8·28-s − 6·29-s + 2·31-s + 12·35-s − 4·37-s + 3·41-s − 7·43-s − 6·44-s + 6·47-s + 9·49-s + 2·52-s + 6·53-s − 9·55-s − 6·59-s + 8·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s − 1.51·7-s + 0.904·11-s − 0.277·13-s + 16-s + 0.242·17-s − 0.229·19-s + 1.34·20-s − 1.87·23-s + 4/5·25-s + 1.51·28-s − 1.11·29-s + 0.359·31-s + 2.02·35-s − 0.657·37-s + 0.468·41-s − 1.06·43-s − 0.904·44-s + 0.875·47-s + 9/7·49-s + 0.277·52-s + 0.824·53-s − 1.21·55-s − 0.781·59-s + 1.02·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(153\)    =    \(3^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{153} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 153,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−19.46687249201034, −19.17680587827487, −18.20744413775290, −16.98973965644869, −16.26536499550713, −15.38469688332817, −14.39633194636082, −13.38144935083502, −12.40008400828974, −11.81013235135079, −10.23122640158842, −9.392750473138959, −8.382258919016961, −7.251325541264254, −5.941553212265887, −4.191142628833387, −3.523434152120564, 0, 3.523434152120564, 4.191142628833387, 5.941553212265887, 7.251325541264254, 8.382258919016961, 9.392750473138959, 10.23122640158842, 11.81013235135079, 12.40008400828974, 13.38144935083502, 14.39633194636082, 15.38469688332817, 16.26536499550713, 16.98973965644869, 18.20744413775290, 19.17680587827487, 19.46687249201034

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