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  − 7.80·2-s − 2.71·3-s − 67.0·4-s − 402.·5-s + 21.1·6-s − 356.·7-s + 1.52e3·8-s − 2.17e3·9-s + 3.13e3·10-s + 338.·11-s + 181.·12-s − 1.90e3·13-s + 2.78e3·14-s + 1.09e3·15-s − 3.30e3·16-s − 1.37e4·17-s + 1.70e4·18-s + 3.23e4·19-s + 2.69e4·20-s + 968.·21-s − 2.64e3·22-s + 1.03e5·23-s − 4.13e3·24-s + 8.34e4·25-s + 1.48e4·26-s + 1.18e4·27-s + 2.39e4·28-s + ⋯
L(s)  = 1  − 0.689·2-s − 0.0580·3-s − 0.523·4-s − 1.43·5-s + 0.0400·6-s − 0.393·7-s + 1.05·8-s − 0.996·9-s + 0.992·10-s + 0.0767·11-s + 0.0304·12-s − 0.240·13-s + 0.271·14-s + 0.0834·15-s − 0.201·16-s − 0.676·17-s + 0.687·18-s + 1.08·19-s + 0.753·20-s + 0.0228·21-s − 0.0529·22-s + 1.76·23-s − 0.0610·24-s + 1.06·25-s + 0.165·26-s + 0.115·27-s + 0.205·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(0.452665\)
\(L(\frac12)\)  \(\approx\)  \(0.452665\)
\(L(\frac{9}{2})\)   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 \( 1 + 7.80T + 128T^{2} \)
3 \( 1 + 2.71T + 2.18e3T^{2} \)
5 \( 1 + 402.T + 7.81e4T^{2} \)
7 \( 1 + 356.T + 8.23e5T^{2} \)
11 \( 1 - 338.T + 1.94e7T^{2} \)
13 \( 1 + 1.90e3T + 6.27e7T^{2} \)
17 \( 1 + 1.37e4T + 4.10e8T^{2} \)
19 \( 1 - 3.23e4T + 8.93e8T^{2} \)
23 \( 1 - 1.03e5T + 3.40e9T^{2} \)
29 \( 1 - 97.3T + 1.72e10T^{2} \)
31 \( 1 + 2.91e3T + 2.75e10T^{2} \)
37 \( 1 + 2.80e5T + 9.49e10T^{2} \)
41 \( 1 + 6.80e4T + 1.94e11T^{2} \)
47 \( 1 - 1.45e5T + 5.06e11T^{2} \)
53 \( 1 + 4.50e5T + 1.17e12T^{2} \)
59 \( 1 - 2.16e6T + 2.48e12T^{2} \)
61 \( 1 + 2.40e5T + 3.14e12T^{2} \)
67 \( 1 - 7.63e5T + 6.06e12T^{2} \)
71 \( 1 - 2.56e5T + 9.09e12T^{2} \)
73 \( 1 + 4.25e6T + 1.10e13T^{2} \)
79 \( 1 - 6.60e6T + 1.92e13T^{2} \)
83 \( 1 - 1.57e6T + 2.71e13T^{2} \)
89 \( 1 - 1.01e7T + 4.42e13T^{2} \)
97 \( 1 + 1.90e6T + 8.07e13T^{2} \)
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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

−14.56571582676339453962771164518, −13.23397514502078934662607528022, −11.84994635825642472381815066397, −10.86106503956554879124408322513, −9.268500937091303074358541706051, −8.316925573805999713561220937137, −7.15307244827558591904449538085, −4.95111035611260849295831710547, −3.38232308648273880461724115570, −0.55235583804325382276868960182, 0.55235583804325382276868960182, 3.38232308648273880461724115570, 4.95111035611260849295831710547, 7.15307244827558591904449538085, 8.316925573805999713561220937137, 9.268500937091303074358541706051, 10.86106503956554879124408322513, 11.84994635825642472381815066397, 13.23397514502078934662607528022, 14.56571582676339453962771164518

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