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  + 8.08·2-s − 1.19·3-s − 62.6·4-s + 164.·5-s − 9.67·6-s + 1.31e3·7-s − 1.54e3·8-s − 2.18e3·9-s + 1.33e3·10-s + 8.63e3·11-s + 75.0·12-s + 8.42e3·13-s + 1.06e4·14-s − 197.·15-s − 4.43e3·16-s + 6.30e3·17-s − 1.76e4·18-s + 1.66e4·19-s − 1.03e4·20-s − 1.57e3·21-s + 6.97e4·22-s + 4.41e4·23-s + 1.84e3·24-s − 5.09e4·25-s + 6.80e4·26-s + 5.23e3·27-s − 8.24e4·28-s + ⋯
L(s)  = 1  + 0.714·2-s − 0.0255·3-s − 0.489·4-s + 0.589·5-s − 0.0182·6-s + 1.44·7-s − 1.06·8-s − 0.999·9-s + 0.420·10-s + 1.95·11-s + 0.0125·12-s + 1.06·13-s + 1.03·14-s − 0.0150·15-s − 0.270·16-s + 0.311·17-s − 0.713·18-s + 0.557·19-s − 0.288·20-s − 0.0370·21-s + 1.39·22-s + 0.755·23-s + 0.0272·24-s − 0.652·25-s + 0.759·26-s + 0.0511·27-s − 0.709·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(2.81508\)
\(L(\frac12)\)  \(\approx\)  \(2.81508\)
\(L(\frac{9}{2})\)   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 \( 1 - 8.08T + 128T^{2} \)
3 \( 1 + 1.19T + 2.18e3T^{2} \)
5 \( 1 - 164.T + 7.81e4T^{2} \)
7 \( 1 - 1.31e3T + 8.23e5T^{2} \)
11 \( 1 - 8.63e3T + 1.94e7T^{2} \)
13 \( 1 - 8.42e3T + 6.27e7T^{2} \)
17 \( 1 - 6.30e3T + 4.10e8T^{2} \)
19 \( 1 - 1.66e4T + 8.93e8T^{2} \)
23 \( 1 - 4.41e4T + 3.40e9T^{2} \)
29 \( 1 + 2.38e4T + 1.72e10T^{2} \)
31 \( 1 - 2.71e4T + 2.75e10T^{2} \)
37 \( 1 + 1.14e5T + 9.49e10T^{2} \)
41 \( 1 + 7.92e5T + 1.94e11T^{2} \)
47 \( 1 - 4.90e5T + 5.06e11T^{2} \)
53 \( 1 + 1.64e6T + 1.17e12T^{2} \)
59 \( 1 + 6.25e5T + 2.48e12T^{2} \)
61 \( 1 + 1.22e6T + 3.14e12T^{2} \)
67 \( 1 - 1.75e6T + 6.06e12T^{2} \)
71 \( 1 - 1.15e6T + 9.09e12T^{2} \)
73 \( 1 - 2.26e6T + 1.10e13T^{2} \)
79 \( 1 - 4.91e6T + 1.92e13T^{2} \)
83 \( 1 + 9.34e6T + 2.71e13T^{2} \)
89 \( 1 - 9.44e6T + 4.42e13T^{2} \)
97 \( 1 + 6.08e6T + 8.07e13T^{2} \)
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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

−14.15404003980713116134999121427, −13.76620504009712814254318179990, −12.01094531085254714395820842176, −11.25015228637989101513368871499, −9.297630589975848581218870339974, −8.408883172334507461844504442876, −6.23202116115057725040954717474, −5.09335733977921926787457307257, −3.62744442250063717915336342541, −1.36723945722206286405726657027, 1.36723945722206286405726657027, 3.62744442250063717915336342541, 5.09335733977921926787457307257, 6.23202116115057725040954717474, 8.408883172334507461844504442876, 9.297630589975848581218870339974, 11.25015228637989101513368871499, 12.01094531085254714395820842176, 13.76620504009712814254318179990, 14.15404003980713116134999121427

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