Properties

Degree 2
Conductor $ 3 \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  + 3-s − 2·4-s + 3·5-s − 4·7-s + 9-s − 3·11-s − 2·12-s − 13-s + 3·15-s + 4·16-s − 17-s − 19-s − 6·20-s − 4·21-s + 9·23-s + 4·25-s + 27-s + 8·28-s + 6·29-s + 2·31-s − 3·33-s − 12·35-s − 2·36-s − 4·37-s − 39-s − 3·41-s − 7·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s − 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s + 0.774·15-s + 16-s − 0.242·17-s − 0.229·19-s − 1.34·20-s − 0.872·21-s + 1.87·23-s + 4/5·25-s + 0.192·27-s + 1.51·28-s + 1.11·29-s + 0.359·31-s − 0.522·33-s − 2.02·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s − 0.468·41-s − 1.06·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(51\)    =    \(3 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{51} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 51,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8600590161$
$L(\frac12)$  $\approx$  $0.8600590161$
$L(\frac{3}{2})$   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 \( 1 - T \)
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

−19.49039134625459, −18.75660371646586, −17.73712748235082, −16.77601042028706, −15.42382989853518, −14.11125275639622, −13.19069815363258, −12.84688978513419, −10.23827402727965, −9.642727340101070, −8.618966621189814, −6.664635758267837, −5.143061316475454, −3.006326484375370, 3.006326484375370, 5.143061316475454, 6.664635758267837, 8.618966621189814, 9.642727340101070, 10.23827402727965, 12.84688978513419, 13.19069815363258, 14.11125275639622, 15.42382989853518, 16.77601042028706, 17.73712748235082, 18.75660371646586, 19.49039134625459

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