Properties

Degree 1
Conductor $ 2^{3} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 19-s − 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 35-s − 37-s + 39-s + 41-s − 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯
L(s,χ)  = 1  − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 19-s − 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 35-s − 37-s + 39-s + 41-s − 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{8} (5, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 8,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3736917129$
$L(\frac12,\chi)$  $\approx$  $0.3736917129$
$L(\chi,1)$  $\approx$  0.6232252401
$L(1,\chi)$  $\approx$  0.6232252401

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−49.72903299375465096030823025986, −47.13331010931799950911277678111, −46.53940182717762855601971139271, −44.73172751684639483462528287960, −43.22748486414775009437859200603, −41.34220616442193704510215506015, −39.78688099766664071794355629814, −38.77557777387006581423517488541, −36.54166385177458968906148893903, −34.74577700617002330947269200049, −33.84563089515844186412086967072, −31.63813949132101731704385060746, −29.930764210151905037228608386737, −28.0974449606307973546168663708, −26.9585351803804674196866524131, −24.20196355781560161254720778026, −23.08384999620054654288748795002, −21.13164596222134388263799368067, −18.80595890770714839982375510653, −17.022285974308347338970900304435, −15.19575425064512276844806671236, −12.3105429942365296811289615097, −10.80658816386171201438622675024, −7.62842884176939783416059334365, −4.899973997007036501038304899196, 4.899973997007036501038304899196, 7.62842884176939783416059334365, 10.80658816386171201438622675024, 12.3105429942365296811289615097, 15.19575425064512276844806671236, 17.022285974308347338970900304435, 18.80595890770714839982375510653, 21.13164596222134388263799368067, 23.08384999620054654288748795002, 24.20196355781560161254720778026, 26.9585351803804674196866524131, 28.0974449606307973546168663708, 29.930764210151905037228608386737, 31.63813949132101731704385060746, 33.84563089515844186412086967072, 34.74577700617002330947269200049, 36.54166385177458968906148893903, 38.77557777387006581423517488541, 39.78688099766664071794355629814, 41.34220616442193704510215506015, 43.22748486414775009437859200603, 44.73172751684639483462528287960, 46.53940182717762855601971139271, 47.13331010931799950911277678111, 49.72903299375465096030823025986

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