Properties

Label 2-43-1.1-c7-0-13
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  − 21.7·2-s + 54.2·3-s + 346.·4-s − 409.·5-s − 1.18e3·6-s + 476.·7-s − 4.76e3·8-s + 761.·9-s + 8.93e3·10-s + 6.86e3·11-s + 1.88e4·12-s − 8.27e3·13-s − 1.03e4·14-s − 2.22e4·15-s + 5.94e4·16-s − 1.61e4·17-s − 1.65e4·18-s + 2.75e4·19-s − 1.42e5·20-s + 2.58e4·21-s − 1.49e5·22-s − 6.91e4·23-s − 2.58e5·24-s + 8.99e4·25-s + 1.80e5·26-s − 7.74e4·27-s + 1.65e5·28-s + ⋯
L(s)  = 1  − 1.92·2-s + 1.16·3-s + 2.70·4-s − 1.46·5-s − 2.23·6-s + 0.525·7-s − 3.28·8-s + 0.348·9-s + 2.82·10-s + 1.55·11-s + 3.14·12-s − 1.04·13-s − 1.01·14-s − 1.70·15-s + 3.62·16-s − 0.795·17-s − 0.670·18-s + 0.920·19-s − 3.97·20-s + 0.609·21-s − 2.99·22-s − 1.18·23-s − 3.81·24-s + 1.15·25-s + 2.01·26-s − 0.756·27-s + 1.42·28-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 \( 1 + 21.7T + 128T^{2} \)
3 \( 1 - 54.2T + 2.18e3T^{2} \)
5 \( 1 + 409.T + 7.81e4T^{2} \)
7 \( 1 - 476.T + 8.23e5T^{2} \)
11 \( 1 - 6.86e3T + 1.94e7T^{2} \)
13 \( 1 + 8.27e3T + 6.27e7T^{2} \)
17 \( 1 + 1.61e4T + 4.10e8T^{2} \)
19 \( 1 - 2.75e4T + 8.93e8T^{2} \)
23 \( 1 + 6.91e4T + 3.40e9T^{2} \)
29 \( 1 + 1.11e5T + 1.72e10T^{2} \)
31 \( 1 + 1.69e5T + 2.75e10T^{2} \)
37 \( 1 + 6.01e4T + 9.49e10T^{2} \)
41 \( 1 + 8.13e5T + 1.94e11T^{2} \)
47 \( 1 + 6.24e5T + 5.06e11T^{2} \)
53 \( 1 - 3.65e5T + 1.17e12T^{2} \)
59 \( 1 - 1.43e6T + 2.48e12T^{2} \)
61 \( 1 + 1.53e5T + 3.14e12T^{2} \)
67 \( 1 + 1.27e6T + 6.06e12T^{2} \)
71 \( 1 - 8.21e5T + 9.09e12T^{2} \)
73 \( 1 - 4.08e6T + 1.10e13T^{2} \)
79 \( 1 + 6.74e6T + 1.92e13T^{2} \)
83 \( 1 - 2.81e5T + 2.71e13T^{2} \)
89 \( 1 - 9.40e6T + 4.42e13T^{2} \)
97 \( 1 + 6.78e6T + 8.07e13T^{2} \)
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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

−14.51771442674692958085902909132, −11.92739954796748580547582393398, −11.38190488523409730115706469783, −9.669148657139648966079214350676, −8.737111072649752227342659734028, −7.892463385712967244748099971997, −7.04321258789678050528309888381, −3.55933564106661000678463628052, −1.84616081283351712840215811817, 0, 1.84616081283351712840215811817, 3.55933564106661000678463628052, 7.04321258789678050528309888381, 7.892463385712967244748099971997, 8.737111072649752227342659734028, 9.669148657139648966079214350676, 11.38190488523409730115706469783, 11.92739954796748580547582393398, 14.51771442674692958085902909132

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