Properties

Degree 2
Conductor 43
Sign $0.0858 - 0.996i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (3.02 + 3.79i)2-s + (5.24 + 0.791i)3-s + (−3.46 + 15.1i)4-s + (−13.3 − 9.10i)5-s + (12.8 + 22.3i)6-s + (13.9 − 24.2i)7-s + (−33.0 + 15.9i)8-s + (1.11 + 0.344i)9-s + (−5.86 − 78.1i)10-s + (2.97 + 13.0i)11-s + (−30.1 + 76.8i)12-s + (−3.89 + 51.9i)13-s + (134. − 20.2i)14-s + (−62.8 − 58.3i)15-s + (−48.2 − 23.2i)16-s + (−75.1 + 51.2i)17-s + ⋯
L(s)  = 1  + (1.06 + 1.34i)2-s + (1.00 + 0.152i)3-s + (−0.432 + 1.89i)4-s + (−1.19 − 0.814i)5-s + (0.876 + 1.51i)6-s + (0.754 − 1.30i)7-s + (−1.46 + 0.703i)8-s + (0.0413 + 0.0127i)9-s + (−0.185 − 2.47i)10-s + (0.0816 + 0.357i)11-s + (−0.725 + 1.84i)12-s + (−0.0830 + 1.10i)13-s + (2.56 − 0.386i)14-s + (−1.08 − 1.00i)15-s + (−0.753 − 0.363i)16-s + (−1.07 + 0.731i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0858 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0858 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.0858 - 0.996i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 0.0858 - 0.996i)\)
\(L(2)\)  \(\approx\)  \(1.77839 + 1.63174i\)
\(L(\frac12)\)  \(\approx\)  \(1.77839 + 1.63174i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-281. - 4.77i)T \)
good2 \( 1 + (-3.02 - 3.79i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (-5.24 - 0.791i)T + (25.8 + 7.95i)T^{2} \)
5 \( 1 + (13.3 + 9.10i)T + (45.6 + 116. i)T^{2} \)
7 \( 1 + (-13.9 + 24.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-2.97 - 13.0i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (3.89 - 51.9i)T + (-2.17e3 - 327. i)T^{2} \)
17 \( 1 + (75.1 - 51.2i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (-10.0 + 3.09i)T + (5.66e3 - 3.86e3i)T^{2} \)
23 \( 1 + (-143. + 132. i)T + (909. - 1.21e4i)T^{2} \)
29 \( 1 + (166. - 25.0i)T + (2.33e4 - 7.18e3i)T^{2} \)
31 \( 1 + (-44.7 + 113. i)T + (-2.18e4 - 2.02e4i)T^{2} \)
37 \( 1 + (-132. - 230. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (155. + 195. i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (40.0 - 175. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (4.75 + 63.4i)T + (-1.47e5 + 2.21e4i)T^{2} \)
59 \( 1 + (-26.9 - 12.9i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (-78.6 - 200. i)T + (-1.66e5 + 1.54e5i)T^{2} \)
67 \( 1 + (-15.5 + 4.78i)T + (2.48e5 - 1.69e5i)T^{2} \)
71 \( 1 + (-591. - 548. i)T + (2.67e4 + 3.56e5i)T^{2} \)
73 \( 1 + (-2.26 + 30.2i)T + (-3.84e5 - 5.79e4i)T^{2} \)
79 \( 1 + (-189. + 328. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-600. - 90.4i)T + (5.46e5 + 1.68e5i)T^{2} \)
89 \( 1 + (1.29e3 + 195. i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 + (51.1 + 224. i)T + (-8.22e5 + 3.95e5i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.36600098965642253551070734482, −14.70255029763891482841497144116, −13.78645805300905105593552438924, −12.80368496900801046393496577282, −11.31426649186327132702837264997, −8.822555145727070003228276864442, −7.953415885219209458836447878512, −6.95779544112413718894847941282, −4.54728721190881695312240817506, −4.01591761846647586783700104112, 2.51091359340694453121471179461, 3.43969865992794307288422687341, 5.28197862900283189964808205754, 7.75356358096861489998291508174, 9.101846036259346176350004881061, 11.10839940054667170258599708918, 11.48810320285926328907692523129, 12.79861941726037022483997417908, 13.95467229754615456786861456937, 15.07499694825261873862040088511

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