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  + 18.8·2-s − 90.3·3-s + 227.·4-s + 70.1·5-s − 1.70e3·6-s + 1.06e3·7-s + 1.86e3·8-s + 5.97e3·9-s + 1.32e3·10-s − 1.45e3·11-s − 2.05e4·12-s + 9.74e3·13-s + 2.00e4·14-s − 6.33e3·15-s + 6.11e3·16-s + 2.13e4·17-s + 1.12e5·18-s + 4.52e4·19-s + 1.59e4·20-s − 9.62e4·21-s − 2.74e4·22-s − 3.97e4·23-s − 1.68e5·24-s − 7.32e4·25-s + 1.83e5·26-s − 3.42e5·27-s + 2.42e5·28-s + ⋯
L(s)  = 1  + 1.66·2-s − 1.93·3-s + 1.77·4-s + 0.250·5-s − 3.21·6-s + 1.17·7-s + 1.28·8-s + 2.73·9-s + 0.417·10-s − 0.329·11-s − 3.42·12-s + 1.22·13-s + 1.95·14-s − 0.484·15-s + 0.373·16-s + 1.05·17-s + 4.54·18-s + 1.51·19-s + 0.445·20-s − 2.26·21-s − 0.549·22-s − 0.681·23-s − 2.49·24-s − 0.937·25-s + 2.04·26-s − 3.34·27-s + 2.08·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(3.13255\)
\(L(\frac12)\)  \(\approx\)  \(3.13255\)
\(L(\frac{9}{2})\)   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 \( 1 - 18.8T + 128T^{2} \)
3 \( 1 + 90.3T + 2.18e3T^{2} \)
5 \( 1 - 70.1T + 7.81e4T^{2} \)
7 \( 1 - 1.06e3T + 8.23e5T^{2} \)
11 \( 1 + 1.45e3T + 1.94e7T^{2} \)
13 \( 1 - 9.74e3T + 6.27e7T^{2} \)
17 \( 1 - 2.13e4T + 4.10e8T^{2} \)
19 \( 1 - 4.52e4T + 8.93e8T^{2} \)
23 \( 1 + 3.97e4T + 3.40e9T^{2} \)
29 \( 1 - 1.06e5T + 1.72e10T^{2} \)
31 \( 1 + 1.95e5T + 2.75e10T^{2} \)
37 \( 1 + 3.47e4T + 9.49e10T^{2} \)
41 \( 1 - 2.17e5T + 1.94e11T^{2} \)
47 \( 1 - 9.14e5T + 5.06e11T^{2} \)
53 \( 1 - 1.04e6T + 1.17e12T^{2} \)
59 \( 1 - 2.62e6T + 2.48e12T^{2} \)
61 \( 1 - 3.05e5T + 3.14e12T^{2} \)
67 \( 1 + 1.51e6T + 6.06e12T^{2} \)
71 \( 1 + 3.03e6T + 9.09e12T^{2} \)
73 \( 1 + 4.88e6T + 1.10e13T^{2} \)
79 \( 1 + 2.20e6T + 1.92e13T^{2} \)
83 \( 1 + 8.61e6T + 2.71e13T^{2} \)
89 \( 1 + 4.49e6T + 4.42e13T^{2} \)
97 \( 1 - 1.11e7T + 8.07e13T^{2} \)
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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

−14.21659789214851683244527733886, −13.15676449540398181982942875424, −11.92189045598997508569339696437, −11.47891366057439829751852881919, −10.36008615622823999681744689115, −7.37042351726313548930063800714, −5.81037151623755599716609976222, −5.39135776112932936254692393190, −4.07131043403405661533507681414, −1.35412508978399793859627500443, 1.35412508978399793859627500443, 4.07131043403405661533507681414, 5.39135776112932936254692393190, 5.81037151623755599716609976222, 7.37042351726313548930063800714, 10.36008615622823999681744689115, 11.47891366057439829751852881919, 11.92189045598997508569339696437, 13.15676449540398181982942875424, 14.21659789214851683244527733886

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