Properties

Label 1-993-993.992-r0-0-0
Degree $1$
Conductor $993$
Sign $1$
Analytic cond. $4.61147$
Root an. cond. $4.61147$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 993 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 993 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(993\)    =    \(3 \cdot 331\)
Sign: $1$
Analytic conductor: \(4.61147\)
Root analytic conductor: \(4.61147\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{993} (992, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 993,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.298453969\)
\(L(\frac12)\) \(\approx\) \(2.298453969\)
\(L(1)\) \(\approx\) \(1.632588703\)
\(L(1)\) \(\approx\) \(1.632588703\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
331 \( 1 \)
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 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.09651579295260303834590482857, −20.90168494100649006081262885181, −20.02210896798791955707329603997, −19.469209799631251720286879850672, −19.0917647643996935890222498988, −17.50337606091079620158143381618, −16.74750698544058910323988934366, −15.79934091951732684777436078920, −15.51654228536672549058047029108, −14.49117186685315935726979933659, −13.80996266731870376913348950022, −12.76498869504109557844878763959, −12.19716238760852117781910602010, −11.55936372597722834674274954711, −10.66758249458278981326066102843, −9.627225539182491676221184891, −8.670604256791966654777961800981, −7.349256707525282760102391685687, −6.95129364151737509757438221350, −6.063617069270580501579421734781, −4.840093688952716986516684491795, −4.1910539725255083341821620432, −3.2505479155369903549884766080, −2.574234185465105909101803122360, −0.95905579686796031917785327268, 0.95905579686796031917785327268, 2.574234185465105909101803122360, 3.2505479155369903549884766080, 4.1910539725255083341821620432, 4.840093688952716986516684491795, 6.063617069270580501579421734781, 6.95129364151737509757438221350, 7.349256707525282760102391685687, 8.670604256791966654777961800981, 9.627225539182491676221184891, 10.66758249458278981326066102843, 11.55936372597722834674274954711, 12.19716238760852117781910602010, 12.76498869504109557844878763959, 13.80996266731870376913348950022, 14.49117186685315935726979933659, 15.51654228536672549058047029108, 15.79934091951732684777436078920, 16.74750698544058910323988934366, 17.50337606091079620158143381618, 19.0917647643996935890222498988, 19.469209799631251720286879850672, 20.02210896798791955707329603997, 20.90168494100649006081262885181, 22.09651579295260303834590482857

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