Properties

Degree 1
Conductor $ 2^{3} \cdot 3 \cdot 5 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 29-s + 31-s + 37-s − 41-s + 43-s + 47-s + 49-s − 53-s + 59-s − 61-s + 67-s − 71-s − 73-s − 77-s + 79-s − 83-s − 89-s − 91-s − 97-s + ⋯
L(s,χ)  = 1  − 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 29-s + 31-s + 37-s − 41-s + 43-s + 47-s + 49-s − 53-s + 59-s − 61-s + 67-s − 71-s − 73-s − 77-s + 79-s − 83-s − 89-s − 91-s − 97-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{120} (29, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 120,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.735873224$
$L(\frac12,\chi)$  $\approx$  $1.735873224$
$L(\chi,1)$  $\approx$  1.147147441
$L(1,\chi)$  $\approx$  1.147147441

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

−28.8171372264551480782559072012, −27.87067861474959756546819489167, −26.865525824800293993438651130606, −25.5516551162428526571291856495, −25.16186251807120983569119536429, −23.53776231051619203315229688863, −22.86160558554397493322603808606, −21.74096692365282341174026509101, −20.67989463691621018473514931391, −19.42750683122014713375791919297, −18.79687203256083384587201578980, −17.30380473861409066033331850621, −16.4277349190353886928091180271, −15.34307497206724817155788495461, −14.11645167037890256849191233206, −13.01005586759749042453215746597, −11.976997034497005137370407321631, −10.65728172453007578033307010466, −9.49673843088330743017599881599, −8.43388154894637863228827315, −6.823052292953949953432353696699, −5.96082263611866179835864982702, −4.18962253151872300800023814866, −2.996698971090393673389675147582, −1.03888458426569614280228827842, 1.03888458426569614280228827842, 2.996698971090393673389675147582, 4.18962253151872300800023814866, 5.96082263611866179835864982702, 6.823052292953949953432353696699, 8.43388154894637863228827315, 9.49673843088330743017599881599, 10.65728172453007578033307010466, 11.976997034497005137370407321631, 13.01005586759749042453215746597, 14.11645167037890256849191233206, 15.34307497206724817155788495461, 16.4277349190353886928091180271, 17.30380473861409066033331850621, 18.79687203256083384587201578980, 19.42750683122014713375791919297, 20.67989463691621018473514931391, 21.74096692365282341174026509101, 22.86160558554397493322603808606, 23.53776231051619203315229688863, 25.16186251807120983569119536429, 25.5516551162428526571291856495, 26.865525824800293993438651130606, 27.87067861474959756546819489167, 28.8171372264551480782559072012

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