Properties

Label 1-1723-1723.1722-r1-0-0
Degree $1$
Conductor $1723$
Sign $1$
Analytic cond. $185.162$
Root an. cond. $185.162$
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 − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1723\)
Sign: $1$
Analytic conductor: \(185.162\)
Root analytic conductor: \(185.162\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1723} (1722, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1723,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3915022698\)
\(L(\frac12)\) \(\approx\) \(0.3915022698\)
\(L(1)\) \(\approx\) \(0.3784227507\)
\(L(1)\) \(\approx\) \(0.3784227507\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1723 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
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

−19.949608409306920623009012923893, −19.18401491155709838107346003791, −18.78150907161704501640531484776, −17.819862741887041578364785643374, −17.07385155316963319025957609122, −16.55756175368721514141698772852, −15.90473226292702786545384731260, −15.316959894080823592142367351074, −14.46797916822055020981306513076, −12.84054253315006194632061169116, −12.51331832744588268634057028763, −11.464902624892737697361695217296, −11.2889501173566626691430753316, −10.29662907118026207713618204406, −9.33349839208266606386290558348, −9.0636313064175074284897371219, −7.44123764774914389982966997655, −7.30055880498109754734223163795, −6.45067763133964313491542013499, −5.62145183963049729016120829042, −4.44056016404960703475298862343, −3.57504622099057624731221671621, −2.54193544172015522393207063447, −1.19220513823099894138286389978, −0.368909083139496750303343173473, 0.368909083139496750303343173473, 1.19220513823099894138286389978, 2.54193544172015522393207063447, 3.57504622099057624731221671621, 4.44056016404960703475298862343, 5.62145183963049729016120829042, 6.45067763133964313491542013499, 7.30055880498109754734223163795, 7.44123764774914389982966997655, 9.0636313064175074284897371219, 9.33349839208266606386290558348, 10.29662907118026207713618204406, 11.2889501173566626691430753316, 11.464902624892737697361695217296, 12.51331832744588268634057028763, 12.84054253315006194632061169116, 14.46797916822055020981306513076, 15.316959894080823592142367351074, 15.90473226292702786545384731260, 16.55756175368721514141698772852, 17.07385155316963319025957609122, 17.819862741887041578364785643374, 18.78150907161704501640531484776, 19.18401491155709838107346003791, 19.949608409306920623009012923893

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