Properties

Label 2-43-1.1-c7-0-19
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  + 3.14·2-s + 34.1·3-s − 118.·4-s − 39.2·5-s + 107.·6-s + 434.·7-s − 774.·8-s − 1.02e3·9-s − 123.·10-s − 6.39e3·11-s − 4.02e3·12-s + 442.·13-s + 1.36e3·14-s − 1.34e3·15-s + 1.26e4·16-s − 2.63e4·17-s − 3.22e3·18-s + 7.74e3·19-s + 4.63e3·20-s + 1.48e4·21-s − 2.01e4·22-s − 5.86e4·23-s − 2.64e4·24-s − 7.65e4·25-s + 1.39e3·26-s − 1.09e5·27-s − 5.13e4·28-s + ⋯
L(s)  = 1  + 0.278·2-s + 0.729·3-s − 0.922·4-s − 0.140·5-s + 0.203·6-s + 0.478·7-s − 0.535·8-s − 0.468·9-s − 0.0391·10-s − 1.44·11-s − 0.672·12-s + 0.0559·13-s + 0.133·14-s − 0.102·15-s + 0.773·16-s − 1.29·17-s − 0.130·18-s + 0.259·19-s + 0.129·20-s + 0.349·21-s − 0.403·22-s − 1.00·23-s − 0.390·24-s − 0.980·25-s + 0.0155·26-s − 1.07·27-s − 0.441·28-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 \( 1 - 3.14T + 128T^{2} \)
3 \( 1 - 34.1T + 2.18e3T^{2} \)
5 \( 1 + 39.2T + 7.81e4T^{2} \)
7 \( 1 - 434.T + 8.23e5T^{2} \)
11 \( 1 + 6.39e3T + 1.94e7T^{2} \)
13 \( 1 - 442.T + 6.27e7T^{2} \)
17 \( 1 + 2.63e4T + 4.10e8T^{2} \)
19 \( 1 - 7.74e3T + 8.93e8T^{2} \)
23 \( 1 + 5.86e4T + 3.40e9T^{2} \)
29 \( 1 - 2.70e4T + 1.72e10T^{2} \)
31 \( 1 - 1.00e5T + 2.75e10T^{2} \)
37 \( 1 - 3.58e5T + 9.49e10T^{2} \)
41 \( 1 + 7.43e4T + 1.94e11T^{2} \)
47 \( 1 - 4.62e5T + 5.06e11T^{2} \)
53 \( 1 - 1.17e6T + 1.17e12T^{2} \)
59 \( 1 - 1.70e6T + 2.48e12T^{2} \)
61 \( 1 - 2.64e6T + 3.14e12T^{2} \)
67 \( 1 + 2.11e6T + 6.06e12T^{2} \)
71 \( 1 + 5.06e6T + 9.09e12T^{2} \)
73 \( 1 - 8.99e5T + 1.10e13T^{2} \)
79 \( 1 + 4.05e6T + 1.92e13T^{2} \)
83 \( 1 + 6.07e6T + 2.71e13T^{2} \)
89 \( 1 + 3.93e6T + 4.42e13T^{2} \)
97 \( 1 + 1.24e7T + 8.07e13T^{2} \)
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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

−13.77801665444525337294718964400, −13.11509872435087658902039512180, −11.55813165926833080144549212071, −10.01845026323904839079070772537, −8.661935455338347361789071426322, −7.88397159628716103267500931722, −5.61908934275921559223117112473, −4.21740270220147840353974682427, −2.56332852367118828090026655227, 0, 2.56332852367118828090026655227, 4.21740270220147840353974682427, 5.61908934275921559223117112473, 7.88397159628716103267500931722, 8.661935455338347361789071426322, 10.01845026323904839079070772537, 11.55813165926833080144549212071, 13.11509872435087658902039512180, 13.77801665444525337294718964400

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