Properties

Degree 2
Conductor 43
Sign $-0.998 + 0.0552i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.3 − 16.7i)2-s + (9.91 + 1.49i)3-s + (−74.0 + 324. i)4-s + (247. + 168. i)5-s + (−107. − 186. i)6-s + (−560. + 970. i)7-s + (3.96e3 − 1.90e3i)8-s + (−1.99e3 − 615. i)9-s + (−480. − 6.40e3i)10-s + (−736. − 3.22e3i)11-s + (−1.21e3 + 3.10e3i)12-s + (725. − 9.67e3i)13-s + (2.37e4 − 3.58e3i)14-s + (2.19e3 + 2.04e3i)15-s + (−4.67e4 − 2.24e4i)16-s + (2.01e3 − 1.37e3i)17-s + ⋯
L(s)  = 1  + (−1.18 − 1.48i)2-s + (0.211 + 0.0319i)3-s + (−0.578 + 2.53i)4-s + (0.884 + 0.603i)5-s + (−0.203 − 0.352i)6-s + (−0.617 + 1.06i)7-s + (2.73 − 1.31i)8-s + (−0.911 − 0.281i)9-s + (−0.151 − 2.02i)10-s + (−0.166 − 0.730i)11-s + (−0.203 + 0.519i)12-s + (0.0915 − 1.22i)13-s + (2.31 − 0.349i)14-s + (0.168 + 0.156i)15-s + (−2.85 − 1.37i)16-s + (0.0995 − 0.0678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0552i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.998 + 0.0552i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ -0.998 + 0.0552i)\)
\(L(4)\)  \(\approx\)  \(0.0145222 - 0.525374i\)
\(L(\frac12)\)  \(\approx\)  \(0.0145222 - 0.525374i\)
\(L(\frac{9}{2})\)   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 \( 1 + (6.46e4 + 5.17e5i)T \)
good2 \( 1 + (13.3 + 16.7i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-9.91 - 1.49i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-247. - 168. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (560. - 970. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (736. + 3.22e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (-725. + 9.67e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (-2.01e3 + 1.37e3i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (2.89e4 - 8.94e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (-8.36e4 + 7.75e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (6.57e4 - 9.91e3i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-9.62e4 + 2.45e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-8.43e4 - 1.46e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (6.89e4 + 8.64e4i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (1.02e5 - 4.47e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (6.57e4 + 8.77e5i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (2.67e6 + 1.28e6i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-5.78e5 - 1.47e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (3.26e6 - 1.00e6i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (-4.01e5 - 3.72e5i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (1.74e5 - 2.33e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (-3.18e6 + 5.51e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-5.59e5 - 8.42e4i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (2.04e6 + 3.07e5i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (1.48e6 + 6.51e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−13.36399039168190746686796380533, −12.43904416814909587611724360633, −11.13873193093341626077786860516, −10.22918333044226362944401395328, −9.106428182766676922531980434290, −8.286837621710960940655917284872, −6.05914469419501291434174788314, −3.07457124238940191650941385968, −2.46150241205238158733486584817, −0.32175739334881122266453203767, 1.45101863248771442600505176681, 4.95942776025692694500909708763, 6.37310661807600666940219622903, 7.41492663392357795559816907543, 8.908701713998860862975851026072, 9.558415208574754521234177336888, 10.79467774052968736839482536614, 13.32273939772584111755654178691, 14.07267673406411673137160027051, 15.24282567586030465961789185084

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