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Properties

Label 1-120-120.29-r1-0-0
Degree $1$
Conductor $120$
Sign $1$
Analytic cond. $12.8957$
Root an. cond. $12.8957$
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  − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(12.8957\)
Root analytic conductor: \(12.8957\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{120} (29, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 120,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735873224\)
\(L(\frac12)\) \(\approx\) \(1.735873224\)
\(L(1)\) \(\approx\) \(1.147147441\)
\(L(1)\) \(\approx\) \(1.147147441\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - T \)
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

−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