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Properties

Label 1-2e3-8.5-r0-0-0
Degree $1$
Conductor $8$
Sign $1$
Analytic cond. $0.0371518$
Root an. cond. $0.0371518$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Analytic conductor: \(0.0371518\)
Root analytic conductor: \(0.0371518\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 8,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3736917129\)
\(L(\frac12)\) \(\approx\) \(0.3736917129\)
\(L(1)\) \(\approx\) \(0.6232252401\)
\(L(1)\) \(\approx\) \(0.6232252401\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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