Properties

Degree 2
Conductor $ 3 \cdot 41 $
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 − 2·5-s − 4·7-s + 9-s + 5·11-s + 2·12-s − 4·13-s + 2·15-s + 4·16-s − 5·17-s − 2·19-s + 4·20-s + 4·21-s + 4·23-s − 25-s − 27-s + 8·28-s + 29-s − 5·31-s − 5·33-s + 8·35-s − 2·36-s − 7·37-s + 4·39-s − 41-s + 7·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.50·11-s + 0.577·12-s − 1.10·13-s + 0.516·15-s + 16-s − 1.21·17-s − 0.458·19-s + 0.894·20-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.51·28-s + 0.185·29-s − 0.898·31-s − 0.870·33-s + 1.35·35-s − 1/3·36-s − 1.15·37-s + 0.640·39-s − 0.156·41-s + 1.06·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(123\)    =    \(3 \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{123} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 123,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
41 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.94802771869738, −19.38117369042639, −18.85257314789087, −17.41299591397326, −16.98543383641759, −15.88549583594761, −14.97133832389340, −13.81724582854571, −12.65726821511144, −12.19291964120733, −10.87217891148407, −9.547800136867236, −8.968031421605498, −7.277171208255028, −6.272605169564019, −4.614063463428678, −3.611352503665737, 0, 3.611352503665737, 4.614063463428678, 6.272605169564019, 7.277171208255028, 8.968031421605498, 9.547800136867236, 10.87217891148407, 12.19291964120733, 12.65726821511144, 13.81724582854571, 14.97133832389340, 15.88549583594761, 16.98543383641759, 17.41299591397326, 18.85257314789087, 19.38117369042639, 19.94802771869738

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