Properties

Label 2-43-1.1-c7-0-14
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  − 8.31·2-s + 11.6·3-s − 58.8·4-s + 148.·5-s − 97.2·6-s + 122.·7-s + 1.55e3·8-s − 2.05e3·9-s − 1.23e3·10-s + 5.63e3·11-s − 688.·12-s − 7.73e3·13-s − 1.01e3·14-s + 1.73e3·15-s − 5.38e3·16-s + 1.71e3·17-s + 1.70e4·18-s − 5.11e4·19-s − 8.71e3·20-s + 1.42e3·21-s − 4.68e4·22-s − 7.70e4·23-s + 1.81e4·24-s − 5.61e4·25-s + 6.43e4·26-s − 4.95e4·27-s − 7.19e3·28-s + ⋯
L(s)  = 1  − 0.734·2-s + 0.250·3-s − 0.459·4-s + 0.529·5-s − 0.183·6-s + 0.134·7-s + 1.07·8-s − 0.937·9-s − 0.389·10-s + 1.27·11-s − 0.115·12-s − 0.976·13-s − 0.0989·14-s + 0.132·15-s − 0.328·16-s + 0.0845·17-s + 0.688·18-s − 1.71·19-s − 0.243·20-s + 0.0336·21-s − 0.938·22-s − 1.32·23-s + 0.268·24-s − 0.719·25-s + 0.717·26-s − 0.484·27-s − 0.0619·28-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 \( 1 + 8.31T + 128T^{2} \)
3 \( 1 - 11.6T + 2.18e3T^{2} \)
5 \( 1 - 148.T + 7.81e4T^{2} \)
7 \( 1 - 122.T + 8.23e5T^{2} \)
11 \( 1 - 5.63e3T + 1.94e7T^{2} \)
13 \( 1 + 7.73e3T + 6.27e7T^{2} \)
17 \( 1 - 1.71e3T + 4.10e8T^{2} \)
19 \( 1 + 5.11e4T + 8.93e8T^{2} \)
23 \( 1 + 7.70e4T + 3.40e9T^{2} \)
29 \( 1 + 4.80e4T + 1.72e10T^{2} \)
31 \( 1 + 1.92e5T + 2.75e10T^{2} \)
37 \( 1 - 4.61e5T + 9.49e10T^{2} \)
41 \( 1 - 6.88e5T + 1.94e11T^{2} \)
47 \( 1 + 6.19e5T + 5.06e11T^{2} \)
53 \( 1 - 5.74e5T + 1.17e12T^{2} \)
59 \( 1 + 1.01e6T + 2.48e12T^{2} \)
61 \( 1 + 3.23e6T + 3.14e12T^{2} \)
67 \( 1 - 3.15e6T + 6.06e12T^{2} \)
71 \( 1 - 2.46e6T + 9.09e12T^{2} \)
73 \( 1 + 4.02e6T + 1.10e13T^{2} \)
79 \( 1 + 4.63e5T + 1.92e13T^{2} \)
83 \( 1 + 5.49e6T + 2.71e13T^{2} \)
89 \( 1 + 8.18e6T + 4.42e13T^{2} \)
97 \( 1 - 1.20e7T + 8.07e13T^{2} \)
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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

−14.13033880772185873700829118294, −12.74002605642983596377774362916, −11.24921951609667383299749752590, −9.809959691077780670166448569638, −9.004675748527173833010086597253, −7.83708027940230810939653257226, −6.03238728490822145036137611994, −4.22003505495745856128667345440, −1.96459362656104226334755752141, 0, 1.96459362656104226334755752141, 4.22003505495745856128667345440, 6.03238728490822145036137611994, 7.83708027940230810939653257226, 9.004675748527173833010086597253, 9.809959691077780670166448569638, 11.24921951609667383299749752590, 12.74002605642983596377774362916, 14.13033880772185873700829118294

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