Properties

Degree 2
Conductor 43
Sign $-0.423 + 0.905i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.789 + 0.990i)2-s + (−6.14 − 0.926i)3-s + (1.42 − 6.23i)4-s + (−11.5 − 7.85i)5-s + (−3.93 − 6.82i)6-s + (−0.966 + 1.67i)7-s + (16.4 − 7.91i)8-s + (11.1 + 3.44i)9-s + (−1.31 − 17.6i)10-s + (−7.67 − 33.6i)11-s + (−14.5 + 37.0i)12-s + (−3.66 + 48.8i)13-s + (−2.42 + 0.364i)14-s + (63.5 + 58.9i)15-s + (−25.2 − 12.1i)16-s + (40.7 − 27.7i)17-s + ⋯
L(s)  = 1  + (0.279 + 0.350i)2-s + (−1.18 − 0.178i)3-s + (0.177 − 0.779i)4-s + (−1.03 − 0.702i)5-s + (−0.267 − 0.464i)6-s + (−0.0521 + 0.0903i)7-s + (0.726 − 0.349i)8-s + (0.413 + 0.127i)9-s + (−0.0417 − 0.556i)10-s + (−0.210 − 0.921i)11-s + (−0.349 + 0.890i)12-s + (−0.0780 + 1.04i)13-s + (−0.0461 + 0.00696i)14-s + (1.09 + 1.01i)15-s + (−0.395 − 0.190i)16-s + (0.581 − 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.423 + 0.905i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.423 + 0.905i)\)
\(L(2)\)  \(\approx\)  \(0.366142 - 0.575506i\)
\(L(\frac12)\)  \(\approx\)  \(0.366142 - 0.575506i\)
\(L(\frac{5}{2})\)   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 \( 1 + (263. + 101. i)T \)
good2 \( 1 + (-0.789 - 0.990i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (6.14 + 0.926i)T + (25.8 + 7.95i)T^{2} \)
5 \( 1 + (11.5 + 7.85i)T + (45.6 + 116. i)T^{2} \)
7 \( 1 + (0.966 - 1.67i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (7.67 + 33.6i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (3.66 - 48.8i)T + (-2.17e3 - 327. i)T^{2} \)
17 \( 1 + (-40.7 + 27.7i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (58.3 - 17.9i)T + (5.66e3 - 3.86e3i)T^{2} \)
23 \( 1 + (-138. + 128. i)T + (909. - 1.21e4i)T^{2} \)
29 \( 1 + (-10.8 + 1.63i)T + (2.33e4 - 7.18e3i)T^{2} \)
31 \( 1 + (-26.3 + 67.1i)T + (-2.18e4 - 2.02e4i)T^{2} \)
37 \( 1 + (144. + 250. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-32.8 - 41.1i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (-110. + 482. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (-47.9 - 640. i)T + (-1.47e5 + 2.21e4i)T^{2} \)
59 \( 1 + (446. + 215. i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (-248. - 632. i)T + (-1.66e5 + 1.54e5i)T^{2} \)
67 \( 1 + (-698. + 215. i)T + (2.48e5 - 1.69e5i)T^{2} \)
71 \( 1 + (536. + 498. i)T + (2.67e4 + 3.56e5i)T^{2} \)
73 \( 1 + (61.4 - 820. i)T + (-3.84e5 - 5.79e4i)T^{2} \)
79 \( 1 + (40.2 - 69.6i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-222. - 33.5i)T + (5.46e5 + 1.68e5i)T^{2} \)
89 \( 1 + (-738. - 111. i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 + (239. + 1.04e3i)T + (-8.22e5 + 3.95e5i)T^{2} \)
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

−15.31610804437469338280779281498, −14.02946279116000102414985316743, −12.51796342155124941362411421100, −11.54179225988173241194346010525, −10.64726163811545009793239826877, −8.763931629364927398074133284469, −6.99159281603114125125119995705, −5.75098590011740428662806507500, −4.54847157383210207799517367173, −0.58600678794109437446584048894, 3.32500705182058402909154660580, 4.94859281825706697621498939684, 6.90296872894476786177314231528, 8.002232625543136989900567812450, 10.39477412240252267517514537992, 11.24631775675697764400271772760, 12.09408073643077910505796091821, 13.03700612468315555047956685231, 14.96338377373373227523410431255, 15.84076858475534750103313624811

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