Properties

Degree 2
Conductor 179
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 3·5-s − 4·7-s − 3·9-s + 6·10-s + 4·11-s − 13-s − 8·14-s − 4·16-s + 17-s − 6·18-s − 3·19-s + 6·20-s + 8·22-s + 6·23-s + 4·25-s − 2·26-s − 8·28-s + 3·29-s − 8·31-s − 8·32-s + 2·34-s − 12·35-s − 6·36-s + 2·37-s − 6·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s − 1.51·7-s − 9-s + 1.89·10-s + 1.20·11-s − 0.277·13-s − 2.13·14-s − 16-s + 0.242·17-s − 1.41·18-s − 0.688·19-s + 1.34·20-s + 1.70·22-s + 1.25·23-s + 4/5·25-s − 0.392·26-s − 1.51·28-s + 0.557·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s − 2.02·35-s − 36-s + 0.328·37-s − 0.973·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{179} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 179,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.260198254$
$L(\frac12)$  $\approx$  $2.260198254$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 179$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 179$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad179 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.92352283756612, −19.22918108771940, −17.89749475418358, −16.96750895085431, −16.37406461643449, −14.93825030176383, −14.40647376868733, −13.53497549689768, −12.91966077992608, −12.11851077920996, −10.91540134714577, −9.527249149175652, −9.039531184742746, −6.693152291638342, −6.185108637807887, −5.269271252893545, −3.654552249338981, −2.577424334613899, 2.577424334613899, 3.654552249338981, 5.269271252893545, 6.185108637807887, 6.693152291638342, 9.039531184742746, 9.527249149175652, 10.91540134714577, 12.11851077920996, 12.91966077992608, 13.53497549689768, 14.40647376868733, 14.93825030176383, 16.37406461643449, 16.96750895085431, 17.89749475418358, 19.22918108771940, 19.92352283756612

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