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Properties

Label 1-483-483.482-r1-0-0
Degree $1$
Conductor $483$
Sign $1$
Analytic cond. $51.9055$
Root an. cond. $51.9055$
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  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 13-s + 16-s − 17-s + 19-s − 20-s − 22-s + 25-s + 26-s − 29-s − 31-s − 32-s + 34-s − 37-s − 38-s + 40-s + 41-s − 43-s + 44-s + 47-s − 50-s − 52-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 13-s + 16-s − 17-s + 19-s − 20-s − 22-s + 25-s + 26-s − 29-s − 31-s − 32-s + 34-s − 37-s − 38-s + 40-s + 41-s − 43-s + 44-s + 47-s − 50-s − 52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(51.9055\)
Root analytic conductor: \(51.9055\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{483} (482, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7476468491\)
\(L(\frac12)\) \(\approx\) \(0.7476468491\)
\(L(1)\) \(\approx\) \(0.5717896615\)
\(L(1)\) \(\approx\) \(0.5717896615\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
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

−23.94701263420919948510124091812, −22.574941546614731424419309551062, −21.96904475512342823409526795462, −20.59103915894633739963055952289, −19.90884204685092702408005889267, −19.41471622416387847814702592372, −18.50346172785485005571803205803, −17.571582168567804452402971572927, −16.74970666728609930756149239533, −15.985959950654863675297727416392, −15.10981662891120312702402920432, −14.40095718484139677110598883532, −12.83503967338159059122183803948, −11.80835133542202925196409271946, −11.41144797746271493660820619185, −10.291761844324335081384266282775, −9.23961087601338027010354578004, −8.60472629371595676748948684625, −7.317756212172613179305438475725, −7.06556214531706998420612534561, −5.618704553787816064081169236771, −4.22017860087683391298753141607, −3.157838371601395438433063477184, −1.87283737406454920552464004712, −0.53645746755745833103503849211, 0.53645746755745833103503849211, 1.87283737406454920552464004712, 3.157838371601395438433063477184, 4.22017860087683391298753141607, 5.618704553787816064081169236771, 7.06556214531706998420612534561, 7.317756212172613179305438475725, 8.60472629371595676748948684625, 9.23961087601338027010354578004, 10.291761844324335081384266282775, 11.41144797746271493660820619185, 11.80835133542202925196409271946, 12.83503967338159059122183803948, 14.40095718484139677110598883532, 15.10981662891120312702402920432, 15.985959950654863675297727416392, 16.74970666728609930756149239533, 17.571582168567804452402971572927, 18.50346172785485005571803205803, 19.41471622416387847814702592372, 19.90884204685092702408005889267, 20.59103915894633739963055952289, 21.96904475512342823409526795462, 22.574941546614731424419309551062, 23.94701263420919948510124091812

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