Properties

Degree 2
Conductor 43
Sign $-0.916 - 0.399i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (3.27 + 4.10i)2-s + (−9.59 − 1.44i)3-s + (−4.35 + 19.0i)4-s + (4.62 + 3.15i)5-s + (−25.4 − 44.0i)6-s + (0.410 − 0.710i)7-s + (−54.6 + 26.3i)8-s + (64.0 + 19.7i)9-s + (2.19 + 29.3i)10-s + (10.7 + 47.2i)11-s + (69.3 − 176. i)12-s + (0.341 − 4.55i)13-s + (4.25 − 0.641i)14-s + (−39.8 − 36.9i)15-s + (−146. − 70.3i)16-s + (89.5 − 61.0i)17-s + ⋯
L(s)  = 1  + (1.15 + 1.45i)2-s + (−1.84 − 0.278i)3-s + (−0.544 + 2.38i)4-s + (0.413 + 0.282i)5-s + (−1.73 − 3.00i)6-s + (0.0221 − 0.0383i)7-s + (−2.41 + 1.16i)8-s + (2.37 + 0.732i)9-s + (0.0694 + 0.926i)10-s + (0.295 + 1.29i)11-s + (1.66 − 4.24i)12-s + (0.00728 − 0.0972i)13-s + (0.0812 − 0.0122i)14-s + (−0.685 − 0.635i)15-s + (−2.28 − 1.09i)16-s + (1.27 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.916 - 0.399i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.916 - 0.399i)\)
\(L(2)\)  \(\approx\)  \(0.273684 + 1.31350i\)
\(L(\frac12)\)  \(\approx\)  \(0.273684 + 1.31350i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-94.5 + 265. i)T \)
good2 \( 1 + (-3.27 - 4.10i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (9.59 + 1.44i)T + (25.8 + 7.95i)T^{2} \)
5 \( 1 + (-4.62 - 3.15i)T + (45.6 + 116. i)T^{2} \)
7 \( 1 + (-0.410 + 0.710i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-10.7 - 47.2i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (-0.341 + 4.55i)T + (-2.17e3 - 327. i)T^{2} \)
17 \( 1 + (-89.5 + 61.0i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (23.1 - 7.12i)T + (5.66e3 - 3.86e3i)T^{2} \)
23 \( 1 + (-0.872 + 0.809i)T + (909. - 1.21e4i)T^{2} \)
29 \( 1 + (-31.9 + 4.80i)T + (2.33e4 - 7.18e3i)T^{2} \)
31 \( 1 + (21.6 - 55.0i)T + (-2.18e4 - 2.02e4i)T^{2} \)
37 \( 1 + (-176. - 306. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (39.1 + 49.1i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (-47.4 + 207. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (8.92 + 119. i)T + (-1.47e5 + 2.21e4i)T^{2} \)
59 \( 1 + (-330. - 159. i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (-13.4 - 34.2i)T + (-1.66e5 + 1.54e5i)T^{2} \)
67 \( 1 + (-165. + 51.1i)T + (2.48e5 - 1.69e5i)T^{2} \)
71 \( 1 + (673. + 624. i)T + (2.67e4 + 3.56e5i)T^{2} \)
73 \( 1 + (-2.53 + 33.7i)T + (-3.84e5 - 5.79e4i)T^{2} \)
79 \( 1 + (-306. + 530. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-411. - 62.0i)T + (5.46e5 + 1.68e5i)T^{2} \)
89 \( 1 + (911. + 137. i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 + (96.1 + 421. i)T + (-8.22e5 + 3.95e5i)T^{2} \)
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

−16.07212567052564448552879093336, −14.96400729868217234674654939885, −13.69157289403210454955715753370, −12.47345255533770382040522261771, −11.89115384337053517266814562455, −10.08451039792161076962522821907, −7.49099808198351820411227784249, −6.61813636974391116760670183104, −5.58042938366954707334120762664, −4.54237655740066081904099571010, 1.06467142440298260491877641253, 3.93481368639448211856374884345, 5.43154643192434207050081078686, 6.04978601218645780075815083049, 9.656438289934633810119741533844, 10.73316730316765277213065407837, 11.42663502413946903585042970967, 12.39254559370178178009722871609, 13.24068887379156646344666325290, 14.64795133048332434696934269749

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