Properties

Label 2-23-1.1-c11-0-1
Degree $2$
Conductor $23$
Sign $1$
Analytic cond. $17.6718$
Root an. cond. $4.20379$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 43.0·2-s + 78.8·3-s − 194.·4-s − 1.12e4·5-s − 3.39e3·6-s − 9.28e3·7-s + 9.65e4·8-s − 1.70e5·9-s + 4.85e5·10-s − 7.75e5·11-s − 1.53e4·12-s − 1.35e6·13-s + 3.99e5·14-s − 8.89e5·15-s − 3.75e6·16-s + 5.04e6·17-s + 7.35e6·18-s + 1.10e7·19-s + 2.18e6·20-s − 7.32e5·21-s + 3.34e7·22-s − 6.43e6·23-s + 7.61e6·24-s + 7.83e7·25-s + 5.84e7·26-s − 2.74e7·27-s + 1.80e6·28-s + ⋯
L(s)  = 1  − 0.951·2-s + 0.187·3-s − 0.0947·4-s − 1.61·5-s − 0.178·6-s − 0.208·7-s + 1.04·8-s − 0.964·9-s + 1.53·10-s − 1.45·11-s − 0.0177·12-s − 1.01·13-s + 0.198·14-s − 0.302·15-s − 0.896·16-s + 0.862·17-s + 0.918·18-s + 1.02·19-s + 0.152·20-s − 0.0391·21-s + 1.38·22-s − 0.208·23-s + 0.195·24-s + 1.60·25-s + 0.964·26-s − 0.368·27-s + 0.0197·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $1$
Analytic conductor: \(17.6718\)
Root analytic conductor: \(4.20379\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.2954160734\)
\(L(\frac12)\) \(\approx\) \(0.2954160734\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + 6.43e6T \)
good2 \( 1 + 43.0T + 2.04e3T^{2} \)
3 \( 1 - 78.8T + 1.77e5T^{2} \)
5 \( 1 + 1.12e4T + 4.88e7T^{2} \)
7 \( 1 + 9.28e3T + 1.97e9T^{2} \)
11 \( 1 + 7.75e5T + 2.85e11T^{2} \)
13 \( 1 + 1.35e6T + 1.79e12T^{2} \)
17 \( 1 - 5.04e6T + 3.42e13T^{2} \)
19 \( 1 - 1.10e7T + 1.16e14T^{2} \)
29 \( 1 + 3.22e7T + 1.22e16T^{2} \)
31 \( 1 - 2.54e8T + 2.54e16T^{2} \)
37 \( 1 + 3.12e8T + 1.77e17T^{2} \)
41 \( 1 - 8.65e8T + 5.50e17T^{2} \)
43 \( 1 - 7.75e8T + 9.29e17T^{2} \)
47 \( 1 + 2.34e9T + 2.47e18T^{2} \)
53 \( 1 + 4.77e9T + 9.26e18T^{2} \)
59 \( 1 + 1.57e9T + 3.01e19T^{2} \)
61 \( 1 + 9.63e9T + 4.35e19T^{2} \)
67 \( 1 - 1.06e10T + 1.22e20T^{2} \)
71 \( 1 - 1.06e10T + 2.31e20T^{2} \)
73 \( 1 - 1.53e9T + 3.13e20T^{2} \)
79 \( 1 - 2.42e10T + 7.47e20T^{2} \)
83 \( 1 - 1.34e10T + 1.28e21T^{2} \)
89 \( 1 + 4.54e10T + 2.77e21T^{2} \)
97 \( 1 + 9.36e10T + 7.15e21T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.55334368322934735163214931894, −14.10759521342452228086202750440, −12.41388508406357045337550549372, −11.12038943879527006722490321698, −9.724102876394529503030005242495, −8.111210586299896866854035188583, −7.65912790135869598503074910408, −4.91638655469529011677625270946, −3.06737697485531784348868406772, −0.42104002175365072270312971029, 0.42104002175365072270312971029, 3.06737697485531784348868406772, 4.91638655469529011677625270946, 7.65912790135869598503074910408, 8.111210586299896866854035188583, 9.724102876394529503030005242495, 11.12038943879527006722490321698, 12.41388508406357045337550549372, 14.10759521342452228086202750440, 15.55334368322934735163214931894

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