Properties

Degree 2
Conductor $ 3 \cdot 43 $
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 − 2·7-s + 9-s − 5·11-s + 2·12-s + 3·13-s + 2·15-s + 4·16-s − 3·17-s + 2·19-s + 4·20-s + 2·21-s − 23-s − 25-s − 27-s + 4·28-s − 5·31-s + 5·33-s + 4·35-s − 2·36-s + 8·37-s − 3·39-s − 7·41-s − 43-s + 10·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 0.832·13-s + 0.516·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.894·20-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.755·28-s − 0.898·31-s + 0.870·33-s + 0.676·35-s − 1/3·36-s + 1.31·37-s − 0.480·39-s − 1.09·41-s − 0.152·43-s + 1.50·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(129\)    =    \(3 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{129} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 129,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;43\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−19.17328149808898, −18.33241248113210, −17.87293680186763, −16.44648227366375, −15.89660010660651, −14.94844261538941, −13.34236910234037, −13.08039350791038, −11.83605837473411, −10.75004043596006, −9.761758139217697, −8.510537086333409, −7.516725612687639, −5.963389231448741, −4.743876224277604, −3.477763193362657, 0, 3.477763193362657, 4.743876224277604, 5.963389231448741, 7.516725612687639, 8.510537086333409, 9.761758139217697, 10.75004043596006, 11.83605837473411, 13.08039350791038, 13.34236910234037, 14.94844261538941, 15.89660010660651, 16.44648227366375, 17.87293680186763, 18.33241248113210, 19.17328149808898

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