Properties

Degree 2
Conductor $ 3^{2} \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  + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s + 3·11-s − 5·13-s − 4·14-s − 4·16-s + 17-s − 19-s + 2·20-s + 6·22-s + 7·23-s − 4·25-s − 10·26-s − 4·28-s − 6·29-s + 4·31-s − 8·32-s + 2·34-s − 2·35-s + 10·37-s − 2·38-s − 9·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s + 0.904·11-s − 1.38·13-s − 1.06·14-s − 16-s + 0.242·17-s − 0.229·19-s + 0.447·20-s + 1.27·22-s + 1.45·23-s − 4/5·25-s − 1.96·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 1.41·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.324·38-s − 1.40·41-s + 0.152·43-s + 0.904·44-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  =  0
Selberg data  =  $(2,\ 153,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.076797451$
$L(\frac12)$  $\approx$  $2.076797451$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.56156368884928, −18.63541233621316, −17.25208794239140, −16.70251387066791, −15.32405805060394, −14.77860517602000, −13.84181298106102, −13.04673624974979, −12.28350396167250, −11.40917261844359, −9.902853599394833, −9.106190491838797, −7.216041216492764, −6.254629479625421, −5.198123531256818, −3.955905739677653, −2.623978132742433, 2.623978132742433, 3.955905739677653, 5.198123531256818, 6.254629479625421, 7.216041216492764, 9.106190491838797, 9.902853599394833, 11.40917261844359, 12.28350396167250, 13.04673624974979, 13.84181298106102, 14.77860517602000, 15.32405805060394, 16.70251387066791, 17.25208794239140, 18.63541233621316, 19.56156368884928

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