Properties

Degree 2
Conductor $ 2 \cdot 79 $
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 − 3·3-s + 4-s − 3·5-s − 3·6-s − 3·7-s + 8-s + 6·9-s − 3·10-s − 2·11-s − 3·12-s − 5·13-s − 3·14-s + 9·15-s + 16-s + 6·17-s + 6·18-s − 3·20-s + 9·21-s − 2·22-s − 2·23-s − 3·24-s + 4·25-s − 5·26-s − 9·27-s − 3·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s − 1.22·6-s − 1.13·7-s + 0.353·8-s + 2·9-s − 0.948·10-s − 0.603·11-s − 0.866·12-s − 1.38·13-s − 0.801·14-s + 2.32·15-s + 1/4·16-s + 1.45·17-s + 1.41·18-s − 0.670·20-s + 1.96·21-s − 0.426·22-s − 0.417·23-s − 0.612·24-s + 4/5·25-s − 0.980·26-s − 1.73·27-s − 0.566·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(158\)    =    \(2 \cdot 79\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{158} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 158,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
79 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−19.42673384624898, −19.05690905868863, −17.79443546229921, −16.70842607917921, −16.15052562377458, −15.62231183665732, −14.44641086889521, −12.77246262737706, −12.34878022289235, −11.76229819783504, −10.67275650162719, −9.871463690263606, −7.661383064654329, −6.921161495255821, −5.699246712759102, −4.769324319088354, −3.435435306460354, 0, 3.435435306460354, 4.769324319088354, 5.699246712759102, 6.921161495255821, 7.661383064654329, 9.871463690263606, 10.67275650162719, 11.76229819783504, 12.34878022289235, 12.77246262737706, 14.44641086889521, 15.62231183665732, 16.15052562377458, 16.70842607917921, 17.79443546229921, 19.05690905868863, 19.42673384624898

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