Properties

Degree 2
Conductor 139
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-s + 2·3-s − 4-s − 5-s + 2·6-s + 3·7-s − 3·8-s + 9-s − 10-s + 5·11-s − 2·12-s − 7·13-s + 3·14-s − 2·15-s − 16-s − 6·17-s + 18-s − 2·19-s + 20-s + 6·21-s + 5·22-s + 2·23-s − 6·24-s − 4·25-s − 7·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.577·12-s − 1.94·13-s + 0.801·14-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 1.30·21-s + 1.06·22-s + 0.417·23-s − 1.22·24-s − 4/5·25-s − 1.37·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{139} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 139,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.739686976$
$L(\frac12)$  $\approx$  $1.739686976$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 139$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 139$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad139 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−19.63449062705489, −19.14153303562337, −17.47935847124381, −17.36551069448380, −15.34221325513205, −14.83331263708141, −14.20431114516608, −13.54769088539940, −12.16406954599298, −11.60131157750580, −9.808189396232141, −8.811884136384315, −8.146982879470192, −6.728006103026403, −4.819148607057143, −4.111855597901721, −2.511946114831835, 2.511946114831835, 4.111855597901721, 4.819148607057143, 6.728006103026403, 8.146982879470192, 8.811884136384315, 9.808189396232141, 11.60131157750580, 12.16406954599298, 13.54769088539940, 14.20431114516608, 14.83331263708141, 15.34221325513205, 17.36551069448380, 17.47935847124381, 19.14153303562337, 19.63449062705489

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