Properties

Degree 2
Conductor $ 3^{2} \cdot 13 $
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-s − 4-s − 2·5-s − 4·7-s + 3·8-s + 2·10-s − 4·11-s + 13-s + 4·14-s − 16-s − 2·17-s + 2·20-s + 4·22-s − 25-s − 26-s + 4·28-s + 10·29-s + 4·31-s − 5·32-s + 2·34-s + 8·35-s − 2·37-s − 6·40-s − 6·41-s − 12·43-s + 4·44-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.196·26-s + 0.755·28-s + 1.85·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.328·37-s − 0.948·40-s − 0.937·41-s − 1.82·43-s + 0.603·44-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{117} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 117,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.48437492958482, −18.77631375010964, −18.02004529561296, −16.87513328872506, −15.91826706026988, −15.48759771899207, −13.74514084673726, −13.08852185820390, −12.03518823883712, −10.56874363227147, −9.880880528421930, −8.676495022208841, −7.808468448793005, −6.523008072596238, −4.739281859116205, −3.258028049219911, 0, 3.258028049219911, 4.739281859116205, 6.523008072596238, 7.808468448793005, 8.676495022208841, 9.880880528421930, 10.56874363227147, 12.03518823883712, 13.08852185820390, 13.74514084673726, 15.48759771899207, 15.91826706026988, 16.87513328872506, 18.02004529561296, 18.77631375010964, 19.48437492958482

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