Properties

Label 2-43-43.11-c3-0-9
Degree $2$
Conductor $43$
Sign $-0.318 + 0.947i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.873 − 1.09i)2-s + (4.80 − 6.02i)3-s + (1.34 − 5.88i)4-s + (−14.6 + 7.06i)5-s − 10.8·6-s + 16.8·7-s + (−17.7 + 8.53i)8-s + (−7.22 − 31.6i)9-s + (20.5 + 9.90i)10-s + (−3.70 − 16.2i)11-s + (−29.0 − 36.3i)12-s + (72.4 − 34.8i)13-s + (−14.7 − 18.4i)14-s + (−27.9 + 122. i)15-s + (−18.6 − 8.99i)16-s + (77.8 + 37.5i)17-s + ⋯
L(s)  = 1  + (−0.308 − 0.387i)2-s + (0.925 − 1.16i)3-s + (0.167 − 0.735i)4-s + (−1.31 + 0.632i)5-s − 0.735·6-s + 0.908·7-s + (−0.783 + 0.377i)8-s + (−0.267 − 1.17i)9-s + (0.650 + 0.313i)10-s + (−0.101 − 0.444i)11-s + (−0.698 − 0.875i)12-s + (1.54 − 0.744i)13-s + (−0.280 − 0.351i)14-s + (−0.481 + 2.10i)15-s + (−0.291 − 0.140i)16-s + (1.11 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.318 + 0.947i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.318 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.785563 - 1.09245i\)
\(L(\frac12)\) \(\approx\) \(0.785563 - 1.09245i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (48.3 - 277. i)T \)
good2 \( 1 + (0.873 + 1.09i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (-4.80 + 6.02i)T + (-6.00 - 26.3i)T^{2} \)
5 \( 1 + (14.6 - 7.06i)T + (77.9 - 97.7i)T^{2} \)
7 \( 1 - 16.8T + 343T^{2} \)
11 \( 1 + (3.70 + 16.2i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (-72.4 + 34.8i)T + (1.36e3 - 1.71e3i)T^{2} \)
17 \( 1 + (-77.8 - 37.5i)T + (3.06e3 + 3.84e3i)T^{2} \)
19 \( 1 + (29.9 - 131. i)T + (-6.17e3 - 2.97e3i)T^{2} \)
23 \( 1 + (20.6 + 90.4i)T + (-1.09e4 + 5.27e3i)T^{2} \)
29 \( 1 + (-30.9 - 38.7i)T + (-5.42e3 + 2.37e4i)T^{2} \)
31 \( 1 + (37.2 + 46.7i)T + (-6.62e3 + 2.90e4i)T^{2} \)
37 \( 1 + 128.T + 5.06e4T^{2} \)
41 \( 1 + (-152. - 191. i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (17.3 - 75.8i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (400. + 192. i)T + (9.28e4 + 1.16e5i)T^{2} \)
59 \( 1 + (485. + 233. i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (-391. + 490. i)T + (-5.05e4 - 2.21e5i)T^{2} \)
67 \( 1 + (121. - 533. i)T + (-2.70e5 - 1.30e5i)T^{2} \)
71 \( 1 + (50.5 - 221. i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (663. - 319. i)T + (2.42e5 - 3.04e5i)T^{2} \)
79 \( 1 - 440.T + 4.93e5T^{2} \)
83 \( 1 + (53.2 - 66.8i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (-550. + 689. i)T + (-1.56e5 - 6.87e5i)T^{2} \)
97 \( 1 + (-1.55 - 6.81i)T + (-8.22e5 + 3.95e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.67429514421890343083089626291, −14.32792221524802306009701032240, −12.66293265127770133909587885425, −11.44923836777170196962662546800, −10.53451167195218689305253079937, −8.316987753672268010605447659385, −7.949370662968995439554438782994, −6.18780171944983480591057405954, −3.33847979965803905854288283618, −1.35735315134370808907838088324, 3.52208445989155807510196968642, 4.56044769699492311988297520121, 7.47208858578504970145808085037, 8.473891500171184668858010294833, 9.126620141402675016636503421263, 11.09740532559193747083569265636, 12.07815917985841002209479226969, 13.75505299254438800294632178536, 15.19682201741115527565994828292, 15.75447394798871305381480793450

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