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  − 20.3·2-s − 17.6·3-s + 287.·4-s + 405.·5-s + 360.·6-s + 52.5·7-s − 3.24e3·8-s − 1.87e3·9-s − 8.25e3·10-s + 2.54e3·11-s − 5.08e3·12-s + 7.13e3·13-s − 1.07e3·14-s − 7.16e3·15-s + 2.93e4·16-s − 2.25e4·17-s + 3.81e4·18-s − 6.39e3·19-s + 1.16e5·20-s − 930.·21-s − 5.18e4·22-s + 9.98e4·23-s + 5.74e4·24-s + 8.59e4·25-s − 1.45e5·26-s + 7.18e4·27-s + 1.51e4·28-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.378·3-s + 2.24·4-s + 1.44·5-s + 0.681·6-s + 0.0579·7-s − 2.24·8-s − 0.856·9-s − 2.61·10-s + 0.576·11-s − 0.849·12-s + 0.901·13-s − 0.104·14-s − 0.548·15-s + 1.79·16-s − 1.11·17-s + 1.54·18-s − 0.213·19-s + 3.25·20-s − 0.0219·21-s − 1.03·22-s + 1.71·23-s + 0.847·24-s + 1.10·25-s − 1.62·26-s + 0.702·27-s + 0.130·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(0.862151\)
\(L(\frac12)\)  \(\approx\)  \(0.862151\)
\(L(\frac{9}{2})\)   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 \( 1 + 20.3T + 128T^{2} \)
3 \( 1 + 17.6T + 2.18e3T^{2} \)
5 \( 1 - 405.T + 7.81e4T^{2} \)
7 \( 1 - 52.5T + 8.23e5T^{2} \)
11 \( 1 - 2.54e3T + 1.94e7T^{2} \)
13 \( 1 - 7.13e3T + 6.27e7T^{2} \)
17 \( 1 + 2.25e4T + 4.10e8T^{2} \)
19 \( 1 + 6.39e3T + 8.93e8T^{2} \)
23 \( 1 - 9.98e4T + 3.40e9T^{2} \)
29 \( 1 - 5.60e3T + 1.72e10T^{2} \)
31 \( 1 + 1.63e5T + 2.75e10T^{2} \)
37 \( 1 - 1.98e5T + 9.49e10T^{2} \)
41 \( 1 - 7.42e5T + 1.94e11T^{2} \)
47 \( 1 - 7.86e5T + 5.06e11T^{2} \)
53 \( 1 - 2.08e6T + 1.17e12T^{2} \)
59 \( 1 - 6.07e5T + 2.48e12T^{2} \)
61 \( 1 - 2.59e6T + 3.14e12T^{2} \)
67 \( 1 + 7.55e5T + 6.06e12T^{2} \)
71 \( 1 + 3.83e5T + 9.09e12T^{2} \)
73 \( 1 - 2.94e6T + 1.10e13T^{2} \)
79 \( 1 + 7.28e6T + 1.92e13T^{2} \)
83 \( 1 + 1.91e6T + 2.71e13T^{2} \)
89 \( 1 - 1.05e7T + 4.42e13T^{2} \)
97 \( 1 - 1.18e7T + 8.07e13T^{2} \)
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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

−14.67939024778698449108481399990, −13.20314077780644101741575506962, −11.35148996945348348078231339603, −10.67467669753894004140207873455, −9.292510345633877497285643063693, −8.719009546656650548725557439980, −6.82878118393963187241332985765, −5.82676303008916975995440819027, −2.36711558497486924265681966520, −0.957960980776983754996256738882, 0.957960980776983754996256738882, 2.36711558497486924265681966520, 5.82676303008916975995440819027, 6.82878118393963187241332985765, 8.719009546656650548725557439980, 9.292510345633877497285643063693, 10.67467669753894004140207873455, 11.35148996945348348078231339603, 13.20314077780644101741575506962, 14.67939024778698449108481399990

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