Properties

Degree 2
Conductor 43
Sign $0.747 - 0.664i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.113 + 0.497i)2-s + (4.85 + 4.50i)3-s + (6.97 − 3.35i)4-s + (−8.91 + 1.34i)5-s + (−1.69 + 2.92i)6-s + (2.21 + 3.83i)7-s + (5.00 + 6.27i)8-s + (1.26 + 16.9i)9-s + (−1.68 − 4.28i)10-s + (5.96 + 2.87i)11-s + (49.0 + 15.1i)12-s + (4.94 − 12.6i)13-s + (−1.65 + 1.53i)14-s + (−49.3 − 33.6i)15-s + (36.0 − 45.2i)16-s + (−125. − 18.9i)17-s + ⋯
L(s)  = 1  + (0.0401 + 0.175i)2-s + (0.935 + 0.867i)3-s + (0.871 − 0.419i)4-s + (−0.797 + 0.120i)5-s + (−0.115 + 0.199i)6-s + (0.119 + 0.207i)7-s + (0.221 + 0.277i)8-s + (0.0469 + 0.626i)9-s + (−0.0531 − 0.135i)10-s + (0.163 + 0.0787i)11-s + (1.17 + 0.363i)12-s + (0.105 − 0.268i)13-s + (−0.0316 + 0.0293i)14-s + (−0.850 − 0.579i)15-s + (0.563 − 0.706i)16-s + (−1.79 − 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.747 - 0.664i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 0.747 - 0.664i)\)
\(L(2)\)  \(\approx\)  \(1.68689 + 0.641700i\)
\(L(\frac12)\)  \(\approx\)  \(1.68689 + 0.641700i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-211. - 186. i)T \)
good2 \( 1 + (-0.113 - 0.497i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (-4.85 - 4.50i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (8.91 - 1.34i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (-2.21 - 3.83i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-5.96 - 2.87i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (-4.94 + 12.6i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (125. + 18.9i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (-6.32 + 84.3i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-10.5 + 7.17i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (-60.8 + 56.4i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (-62.7 - 19.3i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (130. - 226. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-54.5 - 238. i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (513. - 247. i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (-174. - 444. i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (-55.0 + 69.0i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (272. - 83.9i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (46.2 - 617. i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (13.0 + 8.87i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (-421. + 1.07e3i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-373. - 647. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (196. + 182. i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (-843. - 782. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (-570. - 274. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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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

−15.39880623641689699520533140037, −14.97484342650836841304837301795, −13.61107668024753163508573617272, −11.75112309430634776329499972518, −10.80910929072027002849720247337, −9.409840984234451444175346657679, −8.151003616325137644637141332182, −6.66920959656302721832007339048, −4.54354322874249461041989926351, −2.79383736435483475549315021709, 2.05846187521241164516872182523, 3.80023420852174418938252288988, 6.70638097804510022952938960461, 7.73006814075107018179336005497, 8.679247942386919244027818265235, 10.79684282505708142982196034022, 11.93949400657852903842593343139, 12.92054945554079668566041196948, 14.05940866257958224744872216881, 15.34275270115517860079663632739

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