Properties

Label 2-23-1.1-c5-0-4
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $3.68882$
Root an. cond. $1.92063$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.15·2-s + 9.09·3-s + 51.7·4-s + 3.86·5-s − 83.2·6-s − 226.·7-s − 180.·8-s − 160.·9-s − 35.3·10-s + 18.3·11-s + 470.·12-s − 711.·13-s + 2.07e3·14-s + 35.1·15-s − 3.60·16-s + 165.·17-s + 1.46e3·18-s + 213.·19-s + 199.·20-s − 2.05e3·21-s − 167.·22-s − 529·23-s − 1.64e3·24-s − 3.11e3·25-s + 6.50e3·26-s − 3.66e3·27-s − 1.17e4·28-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.583·3-s + 1.61·4-s + 0.0691·5-s − 0.943·6-s − 1.74·7-s − 0.997·8-s − 0.659·9-s − 0.111·10-s + 0.0456·11-s + 0.943·12-s − 1.16·13-s + 2.82·14-s + 0.0403·15-s − 0.00352·16-s + 0.138·17-s + 1.06·18-s + 0.135·19-s + 0.111·20-s − 1.01·21-s − 0.0738·22-s − 0.208·23-s − 0.581·24-s − 0.995·25-s + 1.88·26-s − 0.968·27-s − 2.82·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(3.68882\)
Root analytic conductor: \(1.92063\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + 529T \)
good2 \( 1 + 9.15T + 32T^{2} \)
3 \( 1 - 9.09T + 243T^{2} \)
5 \( 1 - 3.86T + 3.12e3T^{2} \)
7 \( 1 + 226.T + 1.68e4T^{2} \)
11 \( 1 - 18.3T + 1.61e5T^{2} \)
13 \( 1 + 711.T + 3.71e5T^{2} \)
17 \( 1 - 165.T + 1.41e6T^{2} \)
19 \( 1 - 213.T + 2.47e6T^{2} \)
29 \( 1 - 5.47e3T + 2.05e7T^{2} \)
31 \( 1 - 6.07e3T + 2.86e7T^{2} \)
37 \( 1 - 1.14e4T + 6.93e7T^{2} \)
41 \( 1 + 1.31e4T + 1.15e8T^{2} \)
43 \( 1 + 2.14e4T + 1.47e8T^{2} \)
47 \( 1 - 1.29e4T + 2.29e8T^{2} \)
53 \( 1 + 2.91e4T + 4.18e8T^{2} \)
59 \( 1 - 1.05e4T + 7.14e8T^{2} \)
61 \( 1 + 3.41e4T + 8.44e8T^{2} \)
67 \( 1 - 1.29e4T + 1.35e9T^{2} \)
71 \( 1 - 4.52e4T + 1.80e9T^{2} \)
73 \( 1 + 5.25e4T + 2.07e9T^{2} \)
79 \( 1 - 5.76e4T + 3.07e9T^{2} \)
83 \( 1 + 4.08e4T + 3.93e9T^{2} \)
89 \( 1 - 5.46e4T + 5.58e9T^{2} \)
97 \( 1 + 1.17e5T + 8.58e9T^{2} \)
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

−16.52216286400100048398235069591, −15.37947389080488218696688756904, −13.64776901001646144077118082941, −11.96117264130566752377549293114, −10.07161753068757523632361854977, −9.426206077462713453649794813770, −8.054824913865107838436730598268, −6.54469224639957625151710339711, −2.77232553239567056013537377254, 0, 2.77232553239567056013537377254, 6.54469224639957625151710339711, 8.054824913865107838436730598268, 9.426206077462713453649794813770, 10.07161753068757523632361854977, 11.96117264130566752377549293114, 13.64776901001646144077118082941, 15.37947389080488218696688756904, 16.52216286400100048398235069591

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