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Properties

Label 1-723-723.722-r1-0-0
Degree $1$
Conductor $723$
Sign $1$
Analytic cond. $77.6971$
Root an. cond. $77.6971$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $1$
Analytic conductor: \(77.6971\)
Root analytic conductor: \(77.6971\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{723} (722, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 723,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4618098119\)
\(L(\frac12)\) \(\approx\) \(0.4618098119\)
\(L(1)\) \(\approx\) \(0.4673483517\)
\(L(1)\) \(\approx\) \(0.4673483517\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 \)
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 \( 1 - T \)
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

−22.3801140479306106439665261291, −21.4729209256039889132065560932, −20.314083180893891270281766768874, −19.74833595013161681890656205449, −19.01875282375358722680616145320, −18.713549821969878327667615109572, −17.11893964991191721120575938389, −16.85595124767436654497346828797, −16.024338638692884454326263059108, −15.03667599251513102686365000720, −14.59394722015952270198362922667, −12.87554393992156189311940837420, −12.2440316283368091439283208376, −11.4878121952881518207622208904, −10.55272304473681005038290413907, −9.63333456601657292214318870884, −8.96127812852369975666219733270, −7.99245807143457710500199889688, −7.05368134361638534773436463989, −6.57519615159612360890896484333, −5.217282183980402878294928800127, −3.73310417936947975199329896831, −3.10439354757593737191662783361, −1.71478879680033717690255415475, −0.38686937177436045985636347702, 0.38686937177436045985636347702, 1.71478879680033717690255415475, 3.10439354757593737191662783361, 3.73310417936947975199329896831, 5.217282183980402878294928800127, 6.57519615159612360890896484333, 7.05368134361638534773436463989, 7.99245807143457710500199889688, 8.96127812852369975666219733270, 9.63333456601657292214318870884, 10.55272304473681005038290413907, 11.4878121952881518207622208904, 12.2440316283368091439283208376, 12.87554393992156189311940837420, 14.59394722015952270198362922667, 15.03667599251513102686365000720, 16.024338638692884454326263059108, 16.85595124767436654497346828797, 17.11893964991191721120575938389, 18.713549821969878327667615109572, 19.01875282375358722680616145320, 19.74833595013161681890656205449, 20.314083180893891270281766768874, 21.4729209256039889132065560932, 22.3801140479306106439665261291

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