Properties

Degree 2
Conductor 43
Sign $-0.984 - 0.175i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.582 − 2.55i)2-s + (−4.15 − 3.85i)3-s + (1.02 − 0.492i)4-s + (−13.6 + 2.05i)5-s + (−7.42 + 12.8i)6-s + (9.34 + 16.1i)7-s + (−14.9 − 18.7i)8-s + (0.380 + 5.07i)9-s + (13.2 + 33.6i)10-s + (−41.3 − 19.9i)11-s + (−6.14 − 1.89i)12-s + (13.3 − 33.9i)13-s + (35.9 − 33.3i)14-s + (64.5 + 44.0i)15-s + (−33.4 + 41.9i)16-s + (62.1 + 9.36i)17-s + ⋯
L(s)  = 1  + (−0.206 − 0.903i)2-s + (−0.799 − 0.741i)3-s + (0.127 − 0.0616i)4-s + (−1.22 + 0.183i)5-s + (−0.504 + 0.874i)6-s + (0.504 + 0.874i)7-s + (−0.659 − 0.827i)8-s + (0.0140 + 0.188i)9-s + (0.417 + 1.06i)10-s + (−1.13 − 0.545i)11-s + (−0.147 − 0.0456i)12-s + (0.284 − 0.724i)13-s + (0.685 − 0.636i)14-s + (1.11 + 0.758i)15-s + (−0.522 + 0.655i)16-s + (0.886 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.984 - 0.175i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.984 - 0.175i)\)
\(L(2)\)  \(\approx\)  \(0.0538159 + 0.609622i\)
\(L(\frac12)\)  \(\approx\)  \(0.0538159 + 0.609622i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-16.1 + 281. i)T \)
good2 \( 1 + (0.582 + 2.55i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (4.15 + 3.85i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (13.6 - 2.05i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (-9.34 - 16.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (41.3 + 19.9i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (-13.3 + 33.9i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (-62.1 - 9.36i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (-11.9 + 159. i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-30.1 + 20.5i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (0.0346 - 0.0321i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (-142. - 43.9i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (157. - 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (3.55 + 15.5i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (-359. + 173. i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (-140. - 356. i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (-326. + 409. i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (-315. + 97.3i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (13.4 - 179. i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (343. + 233. i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (312. - 795. i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (527. + 914. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-43.9 - 40.7i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (1.00e3 + 933. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (712. + 343. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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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

−15.29540786012115499342995030464, −13.09446251155725769710153816671, −12.02272771383544111320623451308, −11.50273500556218469073876058200, −10.54819819964915286450610685474, −8.530613031274806532002090708591, −7.11751206037655239847800305666, −5.51320394113318502373891608767, −2.96322544168378304067210413393, −0.57429896065672115685097170996, 4.14888235372942873660968726123, 5.54795061870650394452019448718, 7.43253506126836834701001592859, 8.079611537465945198289068276696, 10.21707562788721932223581964742, 11.28594775080635695910461402223, 12.20845435985990374644543093012, 14.22379642324075952200623041055, 15.39633375031659341405691704298, 16.25697784595915796711941316408

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