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Properties

Label 1-120-120.59-r0-0-0
Degree $1$
Conductor $120$
Sign $1$
Analytic cond. $0.557277$
Root an. cond. $0.557277$
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

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

Invariants

Degree: \(1\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.557277\)
Root analytic conductor: \(0.557277\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 120,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149215425\)
\(L(\frac12)\) \(\approx\) \(1.149215425\)
\(L(1)\) \(\approx\) \(1.127932330\)
\(L(1)\) \(\approx\) \(1.127932330\)

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

−28.982140207410057996336549133379, −28.074935060237371867615340509014, −27.12884257280050612864021199527, −26.07131959951059403500681375399, −25.08257386833782592466613619016, −23.86564960803566866876616501728, −23.26927481148249688057511875633, −21.79468690441909084574470403376, −20.91337187943566668974419145833, −20.08931751682772219958213649235, −18.44490256568450971683725872267, −18.04743294423351394800614766455, −16.56649937922658902075908214001, −15.59504248417446592868425038840, −14.37577487231514114992074383834, −13.443469545721206740395269238811, −12.07257609669878390484355072981, −11.031335583311830487655088003658, −9.935207988446708578658089462093, −8.38254903214624493630088592620, −7.60584318594698389110270103993, −5.89926284173680207628506591625, −4.81135245562906648591481346564, −3.24122814262078206609780379837, −1.53751177076969737392069083542, 1.53751177076969737392069083542, 3.24122814262078206609780379837, 4.81135245562906648591481346564, 5.89926284173680207628506591625, 7.60584318594698389110270103993, 8.38254903214624493630088592620, 9.935207988446708578658089462093, 11.031335583311830487655088003658, 12.07257609669878390484355072981, 13.443469545721206740395269238811, 14.37577487231514114992074383834, 15.59504248417446592868425038840, 16.56649937922658902075908214001, 18.04743294423351394800614766455, 18.44490256568450971683725872267, 20.08931751682772219958213649235, 20.91337187943566668974419145833, 21.79468690441909084574470403376, 23.26927481148249688057511875633, 23.86564960803566866876616501728, 25.08257386833782592466613619016, 26.07131959951059403500681375399, 27.12884257280050612864021199527, 28.074935060237371867615340509014, 28.982140207410057996336549133379

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