Properties

Label 2-43-43.6-c9-0-9
Degree $2$
Conductor $43$
Sign $0.259 - 0.965i$
Analytic cond. $22.1465$
Root an. cond. $4.70601$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 39.3·2-s + (7.60 − 13.1i)3-s + 1.03e3·4-s + (95.5 − 165. i)5-s + (−299. + 518. i)6-s + (4.79e3 + 8.29e3i)7-s − 2.06e4·8-s + (9.72e3 + 1.68e4i)9-s + (−3.76e3 + 6.51e3i)10-s + 4.43e4·11-s + (7.88e3 − 1.36e4i)12-s + (−6.52e4 − 1.12e5i)13-s + (−1.88e5 − 3.26e5i)14-s + (−1.45e3 − 2.51e3i)15-s + 2.82e5·16-s + (3.12e4 + 5.41e4i)17-s + ⋯
L(s)  = 1  − 1.73·2-s + (0.0541 − 0.0938i)3-s + 2.02·4-s + (0.0683 − 0.118i)5-s + (−0.0942 + 0.163i)6-s + (0.754 + 1.30i)7-s − 1.78·8-s + (0.494 + 0.855i)9-s + (−0.118 + 0.205i)10-s + 0.912·11-s + (0.109 − 0.190i)12-s + (−0.633 − 1.09i)13-s + (−1.31 − 2.27i)14-s + (−0.00740 − 0.0128i)15-s + 1.07·16-s + (0.0908 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(22.1465\)
Root analytic conductor: \(4.70601\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :9/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.717079 + 0.549726i\)
\(L(\frac12)\) \(\approx\) \(0.717079 + 0.549726i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.03e7 - 1.98e7i)T \)
good2 \( 1 + 39.3T + 512T^{2} \)
3 \( 1 + (-7.60 + 13.1i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-95.5 + 165. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-4.79e3 - 8.29e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 - 4.43e4T + 2.35e9T^{2} \)
13 \( 1 + (6.52e4 + 1.12e5i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (-3.12e4 - 5.41e4i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-4.05e4 + 7.01e4i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-4.16e5 + 7.21e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (1.33e6 + 2.31e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (4.79e6 - 8.30e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-1.97e6 + 3.42e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 2.10e7T + 3.27e14T^{2} \)
47 \( 1 - 4.92e7T + 1.11e15T^{2} \)
53 \( 1 + (-1.81e7 + 3.15e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 - 5.81e7T + 8.66e15T^{2} \)
61 \( 1 + (-3.00e7 - 5.21e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (2.89e7 - 5.00e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-1.82e8 - 3.16e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (-3.71e7 - 6.43e7i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (1.03e8 + 1.78e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (2.10e8 - 3.64e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-2.92e8 + 5.06e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 1.65e9T + 7.60e17T^{2} \)
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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

−14.66756336333735992923337938936, −12.64438671577234166983317200018, −11.45705196628106473011069156493, −10.37515306897929493236633441241, −9.113278110720941047203303803724, −8.272989075364638660315686003089, −7.13558460876929160645146307729, −5.32207562588174308623192170190, −2.41951346848174247162080340553, −1.26441456473166244052503473605, 0.63721870417823908167975149683, 1.72232233936990799662541526191, 4.08278522163665213175346081937, 6.76445129119536113510422948918, 7.46799826914391802537472446165, 8.993236094204532475304870376757, 9.842059489262946914514616818539, 10.96598702832195827815834588800, 11.97245047343803040820982715130, 14.00958945650779194234741595678

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