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-s + 4-s − 5-s + 6-s − 3·7-s − 8-s − 2·9-s + 10-s + 4·11-s − 12-s − 7·13-s + 3·14-s + 15-s + 16-s − 4·17-s + 2·18-s − 6·19-s − 20-s + 3·21-s − 4·22-s + 6·23-s + 24-s − 4·25-s + 7·26-s + 5·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.94·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s − 1.37·19-s − 0.223·20-s + 0.654·21-s − 0.852·22-s + 1.25·23-s + 0.204·24-s − 4/5·25-s + 1.37·26-s + 0.962·27-s − 0.566·28-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 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.53906895832326, −19.16256538134935, −17.63633757793022, −17.03604184066977, −16.58989661349845, −15.32175558213185, −14.66138486987342, −13.15232038291300, −12.05634195288323, −11.54480511636342, −10.28684063846693, −9.410151387289790, −8.410553722468252, −6.898176596036655, −6.333791386314499, −4.580108750496760, −2.775542042019986, 0, 2.775542042019986, 4.580108750496760, 6.333791386314499, 6.898176596036655, 8.410553722468252, 9.410151387289790, 10.28684063846693, 11.54480511636342, 12.05634195288323, 13.15232038291300, 14.66138486987342, 15.32175558213185, 16.58989661349845, 17.03604184066977, 17.63633757793022, 19.16256538134935, 19.53906895832326

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