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  + 17.3·2-s + 73.8·3-s + 172.·4-s − 122.·5-s + 1.28e3·6-s + 247.·7-s + 777.·8-s + 3.26e3·9-s − 2.13e3·10-s + 2.50e3·11-s + 1.27e4·12-s + 1.37e3·13-s + 4.28e3·14-s − 9.07e3·15-s − 8.63e3·16-s − 3.20e4·17-s + 5.66e4·18-s + 1.02e4·19-s − 2.12e4·20-s + 1.82e4·21-s + 4.34e4·22-s − 2.18e4·23-s + 5.73e4·24-s − 6.30e4·25-s + 2.38e4·26-s + 7.95e4·27-s + 4.26e4·28-s + ⋯
L(s)  = 1  + 1.53·2-s + 1.57·3-s + 1.35·4-s − 0.439·5-s + 2.42·6-s + 0.272·7-s + 0.536·8-s + 1.49·9-s − 0.674·10-s + 0.567·11-s + 2.13·12-s + 0.173·13-s + 0.417·14-s − 0.694·15-s − 0.527·16-s − 1.58·17-s + 2.28·18-s + 0.342·19-s − 0.593·20-s + 0.429·21-s + 0.869·22-s − 0.374·23-s + 0.847·24-s − 0.806·25-s + 0.266·26-s + 0.778·27-s + 0.367·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(6.01678\)
\(L(\frac12)\)  \(\approx\)  \(6.01678\)
\(L(\frac{9}{2})\)   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 \( 1 - 17.3T + 128T^{2} \)
3 \( 1 - 73.8T + 2.18e3T^{2} \)
5 \( 1 + 122.T + 7.81e4T^{2} \)
7 \( 1 - 247.T + 8.23e5T^{2} \)
11 \( 1 - 2.50e3T + 1.94e7T^{2} \)
13 \( 1 - 1.37e3T + 6.27e7T^{2} \)
17 \( 1 + 3.20e4T + 4.10e8T^{2} \)
19 \( 1 - 1.02e4T + 8.93e8T^{2} \)
23 \( 1 + 2.18e4T + 3.40e9T^{2} \)
29 \( 1 - 1.73e5T + 1.72e10T^{2} \)
31 \( 1 - 1.60e4T + 2.75e10T^{2} \)
37 \( 1 + 7.56e4T + 9.49e10T^{2} \)
41 \( 1 - 4.52e5T + 1.94e11T^{2} \)
47 \( 1 - 1.13e5T + 5.06e11T^{2} \)
53 \( 1 + 1.42e6T + 1.17e12T^{2} \)
59 \( 1 - 1.48e6T + 2.48e12T^{2} \)
61 \( 1 + 2.69e6T + 3.14e12T^{2} \)
67 \( 1 - 2.19e6T + 6.06e12T^{2} \)
71 \( 1 - 4.77e6T + 9.09e12T^{2} \)
73 \( 1 - 4.98e6T + 1.10e13T^{2} \)
79 \( 1 - 3.39e6T + 1.92e13T^{2} \)
83 \( 1 - 8.35e6T + 2.71e13T^{2} \)
89 \( 1 + 7.83e6T + 4.42e13T^{2} \)
97 \( 1 - 6.12e6T + 8.07e13T^{2} \)
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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

−14.17537952117450683505998029843, −13.68046648085650842306426212162, −12.51933690462303138029463699215, −11.27557701141838662590613316987, −9.295012032737493275869982145081, −8.079134091288438434666495034462, −6.57880897046850835067299464098, −4.53476119191405497457225835453, −3.54342324895824396165458645725, −2.22070434964846614755690740934, 2.22070434964846614755690740934, 3.54342324895824396165458645725, 4.53476119191405497457225835453, 6.57880897046850835067299464098, 8.079134091288438434666495034462, 9.295012032737493275869982145081, 11.27557701141838662590613316987, 12.51933690462303138029463699215, 13.68046648085650842306426212162, 14.17537952117450683505998029843

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