Properties

Label 2-43-43.42-c8-0-17
Degree $2$
Conductor $43$
Sign $0.525 - 0.850i$
Analytic cond. $17.5172$
Root an. cond. $4.18536$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24.2i·2-s + 13.8i·3-s − 333.·4-s − 427. i·5-s − 335.·6-s − 2.56e3i·7-s − 1.89e3i·8-s + 6.37e3·9-s + 1.03e4·10-s + 3.29e3·11-s − 4.61e3i·12-s + 1.78e3·13-s + 6.22e4·14-s + 5.90e3·15-s − 3.94e4·16-s + 1.61e5·17-s + ⋯
L(s)  = 1  + 1.51i·2-s + 0.170i·3-s − 1.30·4-s − 0.684i·5-s − 0.258·6-s − 1.06i·7-s − 0.462i·8-s + 0.970·9-s + 1.03·10-s + 0.225·11-s − 0.222i·12-s + 0.0624·13-s + 1.62·14-s + 0.116·15-s − 0.602·16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.69070 + 0.942438i\)
\(L(\frac12)\) \(\approx\) \(1.69070 + 0.942438i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.79e6 + 2.90e6i)T \)
good2 \( 1 - 24.2iT - 256T^{2} \)
3 \( 1 - 13.8iT - 6.56e3T^{2} \)
5 \( 1 + 427. iT - 3.90e5T^{2} \)
7 \( 1 + 2.56e3iT - 5.76e6T^{2} \)
11 \( 1 - 3.29e3T + 2.14e8T^{2} \)
13 \( 1 - 1.78e3T + 8.15e8T^{2} \)
17 \( 1 - 1.61e5T + 6.97e9T^{2} \)
19 \( 1 + 1.88e5iT - 1.69e10T^{2} \)
23 \( 1 + 4.30e5T + 7.83e10T^{2} \)
29 \( 1 - 1.23e6iT - 5.00e11T^{2} \)
31 \( 1 - 8.19e5T + 8.52e11T^{2} \)
37 \( 1 + 1.89e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.24e6T + 7.98e12T^{2} \)
47 \( 1 - 3.12e6T + 2.38e13T^{2} \)
53 \( 1 - 6.47e6T + 6.22e13T^{2} \)
59 \( 1 + 1.77e5T + 1.46e14T^{2} \)
61 \( 1 + 1.30e7iT - 1.91e14T^{2} \)
67 \( 1 + 1.94e7T + 4.06e14T^{2} \)
71 \( 1 + 1.97e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.15e7iT - 8.06e14T^{2} \)
79 \( 1 - 8.29e6T + 1.51e15T^{2} \)
83 \( 1 - 4.28e7T + 2.25e15T^{2} \)
89 \( 1 + 2.70e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.00e8T + 7.83e15T^{2} \)
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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

−14.51274721036125862051821869266, −13.64655828446114191675650024637, −12.38884845289283378322800872506, −10.46205592225736311787421315362, −9.151658496384113886271785713895, −7.76937889854151367362420861387, −6.90210346680321964372905150360, −5.30043444646400366824245508457, −4.11876159361576966072406739611, −0.906506023504202229375683455935, 1.30201666516492542882992296176, 2.61167721560011482251233663844, 3.96870414845807537169645004866, 6.04344095820293799568549495437, 7.929187995691876013428163884333, 9.734338912506677957629054323442, 10.31314802488710758018209530143, 11.99793160795639495492102048110, 12.20721162037831967359768656648, 13.71526302722188978516330805189

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