Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.0·2-s + 41.5·3-s + 130.·4-s + 431.·5-s + 667.·6-s + 218.·7-s + 36.2·8-s − 463.·9-s + 6.94e3·10-s − 7.01e3·11-s + 5.40e3·12-s + 7.01e3·13-s + 3.51e3·14-s + 1.79e4·15-s − 1.60e4·16-s + 3.70e4·17-s − 7.45e3·18-s − 3.98e4·19-s + 5.62e4·20-s + 9.08e3·21-s − 1.12e5·22-s + 2.98e4·23-s + 1.50e3·24-s + 1.08e5·25-s + 1.12e5·26-s − 1.10e5·27-s + 2.85e4·28-s + ⋯
L(s)  = 1  + 1.42·2-s + 0.887·3-s + 1.01·4-s + 1.54·5-s + 1.26·6-s + 0.241·7-s + 0.0250·8-s − 0.212·9-s + 2.19·10-s − 1.58·11-s + 0.903·12-s + 0.885·13-s + 0.342·14-s + 1.37·15-s − 0.982·16-s + 1.82·17-s − 0.301·18-s − 1.33·19-s + 1.57·20-s + 0.214·21-s − 2.25·22-s + 0.511·23-s + 0.0222·24-s + 1.38·25-s + 1.25·26-s − 1.07·27-s + 0.245·28-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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(5.50501\)
\(L(\frac12)\)  \(\approx\)  \(5.50501\)
\(L(\frac{9}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + 7.95e4T \)
good2 \( 1 - 16.0T + 128T^{2} \)
3 \( 1 - 41.5T + 2.18e3T^{2} \)
5 \( 1 - 431.T + 7.81e4T^{2} \)
7 \( 1 - 218.T + 8.23e5T^{2} \)
11 \( 1 + 7.01e3T + 1.94e7T^{2} \)
13 \( 1 - 7.01e3T + 6.27e7T^{2} \)
17 \( 1 - 3.70e4T + 4.10e8T^{2} \)
19 \( 1 + 3.98e4T + 8.93e8T^{2} \)
23 \( 1 - 2.98e4T + 3.40e9T^{2} \)
29 \( 1 + 7.28e4T + 1.72e10T^{2} \)
31 \( 1 + 1.03e5T + 2.75e10T^{2} \)
37 \( 1 - 3.30e5T + 9.49e10T^{2} \)
41 \( 1 - 3.87e5T + 1.94e11T^{2} \)
47 \( 1 + 1.39e6T + 5.06e11T^{2} \)
53 \( 1 + 3.00e5T + 1.17e12T^{2} \)
59 \( 1 + 4.40e5T + 2.48e12T^{2} \)
61 \( 1 - 1.72e6T + 3.14e12T^{2} \)
67 \( 1 - 3.45e6T + 6.06e12T^{2} \)
71 \( 1 + 1.85e6T + 9.09e12T^{2} \)
73 \( 1 + 6.73e5T + 1.10e13T^{2} \)
79 \( 1 - 5.44e6T + 1.92e13T^{2} \)
83 \( 1 - 4.84e6T + 2.71e13T^{2} \)
89 \( 1 - 8.98e6T + 4.42e13T^{2} \)
97 \( 1 + 7.64e6T + 8.07e13T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.35100958550157533427539080633, −13.28909599703985342597474831767, −12.89770399736998542158513968399, −10.95079160563569809625624033258, −9.566315182322169109979443880538, −8.135731658028406805077554792424, −6.06324026070238297280263107045, −5.21801208637145490345045359112, −3.25052740233407094299766576552, −2.14546732053670901113708620622, 2.14546732053670901113708620622, 3.25052740233407094299766576552, 5.21801208637145490345045359112, 6.06324026070238297280263107045, 8.135731658028406805077554792424, 9.566315182322169109979443880538, 10.95079160563569809625624033258, 12.89770399736998542158513968399, 13.28909599703985342597474831767, 14.35100958550157533427539080633

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