Properties

Label 1-420-420.419-r1-0-0
Degree $1$
Conductor $420$
Sign $1$
Analytic cond. $45.1352$
Root an. cond. $45.1352$
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  + 11-s + 13-s − 17-s + 19-s − 23-s − 29-s + 31-s − 37-s + 41-s + 43-s + 47-s + 53-s − 59-s − 61-s + 67-s + 71-s + 73-s − 79-s + 83-s + 89-s + 97-s + ⋯
L(s)  = 1  + 11-s + 13-s − 17-s + 19-s − 23-s − 29-s + 31-s − 37-s + 41-s + 43-s + 47-s + 53-s − 59-s − 61-s + 67-s + 71-s + 73-s − 79-s + 83-s + 89-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(45.1352\)
Root analytic conductor: \(45.1352\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{420} (419, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 420,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.205670957\)
\(L(\frac12)\) \(\approx\) \(2.205670957\)
\(L(1)\) \(\approx\) \(1.226352199\)
\(L(1)\) \(\approx\) \(1.226352199\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + T \)
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

−24.2857137223615323256287812576, −22.93317736013344686151387616174, −22.41849562743010038972889809941, −21.46159061655909404714031460798, −20.43684338250710753206968936119, −19.80918382917924743470378342139, −18.76722497263465231647940853871, −17.91289582577304965782676078951, −17.09905092265090133682795547919, −16.03374932210224468359336509443, −15.386781409253312328258022793583, −14.127647423074157003175734456245, −13.60408608571790769086724164374, −12.38728298427568642183694884254, −11.52876214091407193507601323128, −10.69252338237642040325482302020, −9.47374992571462181413195063102, −8.76463563021733095250440980696, −7.63035142775523711123180845626, −6.54595772057456459323674627453, −5.71974481739633102864661203341, −4.34502133932826795336572652828, −3.51556683738661921119289885292, −2.07588039063739033918877422889, −0.86071547821422477598278446782, 0.86071547821422477598278446782, 2.07588039063739033918877422889, 3.51556683738661921119289885292, 4.34502133932826795336572652828, 5.71974481739633102864661203341, 6.54595772057456459323674627453, 7.63035142775523711123180845626, 8.76463563021733095250440980696, 9.47374992571462181413195063102, 10.69252338237642040325482302020, 11.52876214091407193507601323128, 12.38728298427568642183694884254, 13.60408608571790769086724164374, 14.127647423074157003175734456245, 15.386781409253312328258022793583, 16.03374932210224468359336509443, 17.09905092265090133682795547919, 17.91289582577304965782676078951, 18.76722497263465231647940853871, 19.80918382917924743470378342139, 20.43684338250710753206968936119, 21.46159061655909404714031460798, 22.41849562743010038972889809941, 22.93317736013344686151387616174, 24.2857137223615323256287812576

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