Properties

Degree 2
Conductor 43
Sign $0.523 + 0.852i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 3.18i)2-s + (2.05 + 0.309i)3-s + (3.43 − 15.0i)4-s + (−53.1 − 36.2i)5-s + (4.22 + 7.31i)6-s + (95.1 − 164. i)7-s + (173. − 83.7i)8-s + (−228. − 70.3i)9-s + (−19.5 − 261. i)10-s + (96.9 + 424. i)11-s + (11.6 − 29.8i)12-s + (13.6 − 182. i)13-s + (765. − 115. i)14-s + (−97.8 − 90.8i)15-s + (262. + 126. i)16-s + (1.59e3 − 1.08e3i)17-s + ⋯
L(s)  = 1  + (0.448 + 0.562i)2-s + (0.131 + 0.0198i)3-s + (0.107 − 0.470i)4-s + (−0.951 − 0.648i)5-s + (0.0478 + 0.0829i)6-s + (0.733 − 1.27i)7-s + (0.960 − 0.462i)8-s + (−0.938 − 0.289i)9-s + (−0.0619 − 0.826i)10-s + (0.241 + 1.05i)11-s + (0.0234 − 0.0597i)12-s + (0.0224 − 0.299i)13-s + (1.04 − 0.157i)14-s + (−0.112 − 0.104i)15-s + (0.256 + 0.123i)16-s + (1.33 − 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.523 + 0.852i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.523 + 0.852i)\)
\(L(3)\)  \(\approx\)  \(1.61781 - 0.905026i\)
\(L(\frac12)\)  \(\approx\)  \(1.61781 - 0.905026i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-9.87e3 + 7.04e3i)T \)
good2 \( 1 + (-2.53 - 3.18i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (-2.05 - 0.309i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (53.1 + 36.2i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-95.1 + 164. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-96.9 - 424. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-13.6 + 182. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (-1.59e3 + 1.08e3i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (580. - 179. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (1.03e3 - 963. i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (-2.77e3 + 418. i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (3.01e3 - 7.68e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (1.49e3 + 2.59e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (-1.16e4 - 1.46e4i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (491. - 2.15e3i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (1.32e3 + 1.76e4i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (-4.14e4 - 1.99e4i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (3.30e3 + 8.41e3i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-6.75e4 + 2.08e4i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (-1.05e4 - 9.81e3i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (948. - 1.26e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (3.62e4 - 6.28e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.03e4 - 1.55e3i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (9.30e4 + 1.40e4i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (1.97e4 + 8.66e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.55715885484127098000107320730, −14.09593740896203492814427855926, −12.45746977301810547715671905640, −11.24442512868631694702218617446, −9.910772053949539618736910486766, −8.079334508444901786160744984923, −7.12350238700005288258101525668, −5.21873275230291051921486698418, −4.06693852077435789213579880358, −0.931954902885577058030957567895, 2.49170513999636567288912544723, 3.76908999687189161766090259574, 5.72150872161466285655387746281, 7.85675678620355949719345133324, 8.572991138402195345866448278576, 10.97597717071588990300724061877, 11.56647611551443740209597535596, 12.43718187608700772968193341996, 14.09511630496501269808849237391, 14.85172855635933626295577804201

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