Properties

Label 2-43-1.1-c7-0-22
Degree $2$
Conductor $43$
Sign $-1$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 17.3·2-s − 29.0·3-s + 174.·4-s − 230.·5-s − 505.·6-s − 1.11e3·7-s + 802.·8-s − 1.34e3·9-s − 4.01e3·10-s − 607.·11-s − 5.06e3·12-s + 1.30e4·13-s − 1.93e4·14-s + 6.70e3·15-s − 8.34e3·16-s − 2.06e4·17-s − 2.33e4·18-s − 2.41e4·19-s − 4.01e4·20-s + 3.22e4·21-s − 1.05e4·22-s + 6.03e4·23-s − 2.33e4·24-s − 2.48e4·25-s + 2.26e5·26-s + 1.02e5·27-s − 1.93e5·28-s + ⋯
L(s)  = 1  + 1.53·2-s − 0.621·3-s + 1.36·4-s − 0.825·5-s − 0.954·6-s − 1.22·7-s + 0.554·8-s − 0.613·9-s − 1.26·10-s − 0.137·11-s − 0.845·12-s + 1.64·13-s − 1.88·14-s + 0.513·15-s − 0.509·16-s − 1.01·17-s − 0.942·18-s − 0.808·19-s − 1.12·20-s + 0.760·21-s − 0.211·22-s + 1.03·23-s − 0.344·24-s − 0.318·25-s + 2.52·26-s + 1.00·27-s − 1.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-1$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 - 7.95e4T \)
good2 \( 1 - 17.3T + 128T^{2} \)
3 \( 1 + 29.0T + 2.18e3T^{2} \)
5 \( 1 + 230.T + 7.81e4T^{2} \)
7 \( 1 + 1.11e3T + 8.23e5T^{2} \)
11 \( 1 + 607.T + 1.94e7T^{2} \)
13 \( 1 - 1.30e4T + 6.27e7T^{2} \)
17 \( 1 + 2.06e4T + 4.10e8T^{2} \)
19 \( 1 + 2.41e4T + 8.93e8T^{2} \)
23 \( 1 - 6.03e4T + 3.40e9T^{2} \)
29 \( 1 + 7.77e4T + 1.72e10T^{2} \)
31 \( 1 - 3.46e4T + 2.75e10T^{2} \)
37 \( 1 - 4.89e5T + 9.49e10T^{2} \)
41 \( 1 + 8.09e5T + 1.94e11T^{2} \)
47 \( 1 - 2.89e5T + 5.06e11T^{2} \)
53 \( 1 - 8.92e5T + 1.17e12T^{2} \)
59 \( 1 + 1.51e6T + 2.48e12T^{2} \)
61 \( 1 + 1.60e5T + 3.14e12T^{2} \)
67 \( 1 + 2.42e6T + 6.06e12T^{2} \)
71 \( 1 - 9.93e5T + 9.09e12T^{2} \)
73 \( 1 + 2.67e6T + 1.10e13T^{2} \)
79 \( 1 + 4.46e6T + 1.92e13T^{2} \)
83 \( 1 + 4.10e6T + 2.71e13T^{2} \)
89 \( 1 + 2.96e6T + 4.42e13T^{2} \)
97 \( 1 + 9.83e6T + 8.07e13T^{2} \)
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

−13.52750814324903971000525757458, −12.88884395709901151714318603013, −11.66258313640666679542080161139, −10.91868571740512875919356481756, −8.766442556140550247915642921492, −6.67154862576100893057591115568, −5.85257545093800694652835480014, −4.22252855248802122035494951355, −3.08344799165299122571213337901, 0, 3.08344799165299122571213337901, 4.22252855248802122035494951355, 5.85257545093800694652835480014, 6.67154862576100893057591115568, 8.766442556140550247915642921492, 10.91868571740512875919356481756, 11.66258313640666679542080161139, 12.88884395709901151714318603013, 13.52750814324903971000525757458

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