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Properties

Label 1-1149-1149.1148-r0-0-0
Degree $1$
Conductor $1149$
Sign $1$
Analytic cond. $5.33593$
Root an. cond. $5.33593$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1149\)    =    \(3 \cdot 383\)
Sign: $1$
Analytic conductor: \(5.33593\)
Root analytic conductor: \(5.33593\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1149} (1148, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1149,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389055367\)
\(L(\frac12)\) \(\approx\) \(1.389055367\)
\(L(1)\) \(\approx\) \(0.9725815767\)
\(L(1)\) \(\approx\) \(0.9725815767\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
383 \( 1 \)
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

−21.13667007199285241660263376029, −20.39012944013623034025586518198, −19.8094584778252311839218132947, −18.869500764747415267242824868053, −17.95502424119894951256040393903, −17.50724430207625298871615642152, −17.04087623616266780849854199802, −16.06993622904019501785443341887, −15.09166227909750229451284097579, −14.35263972381306898964872080449, −13.71483536894109271638345903066, −12.36316772660485930062013123564, −11.73073121239301505196478146762, −10.90645283806732256054033648712, −10.06844055630794807240647069772, −9.32162021050206594721201049352, −8.75256184524116337919762610810, −7.675741749072566847526357436949, −6.97556105579087912578401092685, −6.03155188070973358040937780233, −5.21578738778375988198561615180, −4.04557811602007292120792718299, −2.54909295178631738089424494441, −1.94587762644378195201856102079, −1.00678232617289956399280722193, 1.00678232617289956399280722193, 1.94587762644378195201856102079, 2.54909295178631738089424494441, 4.04557811602007292120792718299, 5.21578738778375988198561615180, 6.03155188070973358040937780233, 6.97556105579087912578401092685, 7.675741749072566847526357436949, 8.75256184524116337919762610810, 9.32162021050206594721201049352, 10.06844055630794807240647069772, 10.90645283806732256054033648712, 11.73073121239301505196478146762, 12.36316772660485930062013123564, 13.71483536894109271638345903066, 14.35263972381306898964872080449, 15.09166227909750229451284097579, 16.06993622904019501785443341887, 17.04087623616266780849854199802, 17.50724430207625298871615642152, 17.95502424119894951256040393903, 18.869500764747415267242824868053, 19.8094584778252311839218132947, 20.39012944013623034025586518198, 21.13667007199285241660263376029

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