Properties

Label 2-43-43.9-c7-0-2
Degree $2$
Conductor $43$
Sign $0.991 + 0.128i$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.35 − 7.97i)2-s + (−51.9 − 7.82i)3-s + (5.34 − 23.3i)4-s + (−119. − 81.1i)5-s + (267. + 463. i)6-s + (−639. + 1.10e3i)7-s + (−1.39e3 + 672. i)8-s + (542. + 167. i)9-s + (109. + 1.46e3i)10-s + (−1.89e3 − 8.31e3i)11-s + (−460. + 1.17e3i)12-s + (−777. + 1.03e4i)13-s + (1.28e4 − 1.94e3i)14-s + (5.54e3 + 5.14e3i)15-s + (1.14e4 + 5.52e3i)16-s + (1.35e4 − 9.24e3i)17-s + ⋯
L(s)  = 1  + (−0.562 − 0.704i)2-s + (−1.10 − 0.167i)3-s + (0.0417 − 0.182i)4-s + (−0.426 − 0.290i)5-s + (0.505 + 0.876i)6-s + (−0.704 + 1.22i)7-s + (−0.964 + 0.464i)8-s + (0.248 + 0.0765i)9-s + (0.0347 + 0.463i)10-s + (−0.429 − 1.88i)11-s + (−0.0768 + 0.195i)12-s + (−0.0981 + 1.30i)13-s + (1.25 − 0.189i)14-s + (0.424 + 0.393i)15-s + (0.700 + 0.337i)16-s + (0.669 − 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.991 + 0.128i$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ 0.991 + 0.128i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.360998 - 0.0232695i\)
\(L(\frac12)\) \(\approx\) \(0.360998 - 0.0232695i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (3.11e4 - 5.20e5i)T \)
good2 \( 1 + (6.35 + 7.97i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (51.9 + 7.82i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (119. + 81.1i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (639. - 1.10e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (1.89e3 + 8.31e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (777. - 1.03e4i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (-1.35e4 + 9.24e3i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-1.57e4 + 4.86e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (-3.49e4 + 3.24e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (1.97e5 - 2.97e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-2.56e4 + 6.53e4i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (1.16e4 + 2.02e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-3.43e5 - 4.30e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (2.43e5 - 1.06e6i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (1.34e5 + 1.79e6i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (-1.93e6 - 9.33e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-2.67e5 - 6.80e5i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (1.30e6 - 4.03e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (-3.76e6 - 3.49e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (-1.79e4 + 2.39e5i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (1.48e6 - 2.57e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (4.42e6 + 6.66e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-1.21e5 - 1.82e4i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-1.41e6 - 6.19e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.44737458937280108107994764899, −12.70903192191465166662200652846, −11.48461577482246418449746832412, −11.34716511507367278505147839147, −9.583741518358603968153980676213, −8.584674647945914896031851333500, −6.26577280372635551951756968763, −5.46898873182515054224425094350, −2.86570101252509826152718185772, −0.77178157102912520727253610668, 0.32702267434328959616985154109, 3.58743221143637728037416175932, 5.48402892295914378951286872958, 7.09553407695976552880636279137, 7.61787018746823494340954880048, 9.759539030609257251174934161691, 10.63839499886651739768144718256, 12.12334018161093307979993894211, 13.01326669508571714596446811258, 15.04742383248991517332748741995

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