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Properties

Label 1-159-159.158-r1-0-0
Degree $1$
Conductor $159$
Sign $1$
Analytic cond. $17.0869$
Root an. cond. $17.0869$
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 + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s − 29-s − 31-s + 32-s − 34-s + 35-s + 37-s − 38-s + 40-s + 41-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s − 29-s − 31-s + 32-s − 34-s + 35-s + 37-s − 38-s + 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(159\)    =    \(3 \cdot 53\)
Sign: $1$
Analytic conductor: \(17.0869\)
Root analytic conductor: \(17.0869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159} (158, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 159,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.481355383\)
\(L(\frac12)\) \(\approx\) \(4.481355383\)
\(L(1)\) \(\approx\) \(2.491445035\)
\(L(1)\) \(\approx\) \(2.491445035\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
53 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
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 \( 1 - T \)
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

−27.855989558942335312944577220116, −26.28335460322830807211697218511, −25.5042441997189382626761796682, −24.51093112260310549609632103653, −23.74031744763324748212067363318, −22.74289539576842886421609816581, −21.48505309705567113170525764675, −21.08273055984177211815073200956, −20.21978368626306268667154867645, −18.607666416914181961914921726209, −17.63780791446133822928767854002, −16.541231995206553521211969751918, −15.3051512380369297294098051112, −14.47791796140350096956278303389, −13.35069073815283993864517695531, −12.85747779532554738426375588838, −11.08063274287610338479246173741, −10.75100090398562051078971793566, −8.979780107737490567122818596999, −7.64181300593029919605402538875, −6.28913892546767705772279963659, −5.35904186716410876944024766999, −4.31899836878706193362934768635, −2.633465513250769718284528641567, −1.60165519433427568621382838719, 1.60165519433427568621382838719, 2.633465513250769718284528641567, 4.31899836878706193362934768635, 5.35904186716410876944024766999, 6.28913892546767705772279963659, 7.64181300593029919605402538875, 8.979780107737490567122818596999, 10.75100090398562051078971793566, 11.08063274287610338479246173741, 12.85747779532554738426375588838, 13.35069073815283993864517695531, 14.47791796140350096956278303389, 15.3051512380369297294098051112, 16.541231995206553521211969751918, 17.63780791446133822928767854002, 18.607666416914181961914921726209, 20.21978368626306268667154867645, 21.08273055984177211815073200956, 21.48505309705567113170525764675, 22.74289539576842886421609816581, 23.74031744763324748212067363318, 24.51093112260310549609632103653, 25.5042441997189382626761796682, 26.28335460322830807211697218511, 27.855989558942335312944577220116

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