Properties

Label 2-43-1.1-c7-0-23
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.30·2-s + 47.1·3-s − 58.9·4-s − 187.·5-s + 391.·6-s − 1.50e3·7-s − 1.55e3·8-s + 32.1·9-s − 1.55e3·10-s + 1.94e3·11-s − 2.77e3·12-s − 4.43e3·13-s − 1.24e4·14-s − 8.82e3·15-s − 5.36e3·16-s + 3.18e4·17-s + 266.·18-s + 8.14e3·19-s + 1.10e4·20-s − 7.07e4·21-s + 1.61e4·22-s + 1.19e4·23-s − 7.31e4·24-s − 4.30e4·25-s − 3.68e4·26-s − 1.01e5·27-s + 8.84e4·28-s + ⋯
L(s)  = 1  + 0.734·2-s + 1.00·3-s − 0.460·4-s − 0.670·5-s + 0.739·6-s − 1.65·7-s − 1.07·8-s + 0.0146·9-s − 0.492·10-s + 0.439·11-s − 0.463·12-s − 0.559·13-s − 1.21·14-s − 0.674·15-s − 0.327·16-s + 1.57·17-s + 0.0107·18-s + 0.272·19-s + 0.308·20-s − 1.66·21-s + 0.323·22-s + 0.204·23-s − 1.08·24-s − 0.551·25-s − 0.411·26-s − 0.992·27-s + 0.761·28-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 \( 1 - 8.30T + 128T^{2} \)
3 \( 1 - 47.1T + 2.18e3T^{2} \)
5 \( 1 + 187.T + 7.81e4T^{2} \)
7 \( 1 + 1.50e3T + 8.23e5T^{2} \)
11 \( 1 - 1.94e3T + 1.94e7T^{2} \)
13 \( 1 + 4.43e3T + 6.27e7T^{2} \)
17 \( 1 - 3.18e4T + 4.10e8T^{2} \)
19 \( 1 - 8.14e3T + 8.93e8T^{2} \)
23 \( 1 - 1.19e4T + 3.40e9T^{2} \)
29 \( 1 + 1.15e5T + 1.72e10T^{2} \)
31 \( 1 + 1.00e5T + 2.75e10T^{2} \)
37 \( 1 + 4.55e5T + 9.49e10T^{2} \)
41 \( 1 - 3.96e5T + 1.94e11T^{2} \)
47 \( 1 - 2.49e5T + 5.06e11T^{2} \)
53 \( 1 + 3.63e5T + 1.17e12T^{2} \)
59 \( 1 + 7.05e5T + 2.48e12T^{2} \)
61 \( 1 + 8.31e5T + 3.14e12T^{2} \)
67 \( 1 + 2.27e6T + 6.06e12T^{2} \)
71 \( 1 - 4.30e6T + 9.09e12T^{2} \)
73 \( 1 + 3.27e6T + 1.10e13T^{2} \)
79 \( 1 + 8.41e6T + 1.92e13T^{2} \)
83 \( 1 + 3.46e6T + 2.71e13T^{2} \)
89 \( 1 - 5.16e6T + 4.42e13T^{2} \)
97 \( 1 - 1.54e7T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.93503370326949001947820017399, −12.84723040670952101049391355986, −12.00070727741646381438508375739, −9.799694001702928725475017275066, −8.996012225553624125520334261697, −7.49313556127241329130970529398, −5.75775836673267754748148444061, −3.77114134812089103431181782234, −3.09963917666605238842695124179, 0, 3.09963917666605238842695124179, 3.77114134812089103431181782234, 5.75775836673267754748148444061, 7.49313556127241329130970529398, 8.996012225553624125520334261697, 9.799694001702928725475017275066, 12.00070727741646381438508375739, 12.84723040670952101049391355986, 13.93503370326949001947820017399

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