Properties

Degree 2
Conductor 43
Sign $-0.674 - 0.738i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 − 4.77i)2-s + (−6.48 − 6.01i)3-s + (−14.3 + 6.92i)4-s + (20.4 − 3.07i)5-s + (−21.6 + 37.4i)6-s + (−5.26 − 9.12i)7-s + (24.3 + 30.5i)8-s + (3.82 + 51.0i)9-s + (−36.9 − 94.0i)10-s + (−7.67 − 3.69i)11-s + (134. + 41.6i)12-s + (4.91 − 12.5i)13-s + (−37.8 + 35.0i)14-s + (−150. − 102. i)15-s + (39.4 − 49.4i)16-s + (−46.0 − 6.93i)17-s + ⋯
L(s)  = 1  + (−0.385 − 1.68i)2-s + (−1.24 − 1.15i)3-s + (−1.79 + 0.866i)4-s + (1.82 − 0.275i)5-s + (−1.47 + 2.55i)6-s + (−0.284 − 0.492i)7-s + (1.07 + 1.34i)8-s + (0.141 + 1.88i)9-s + (−1.16 − 2.97i)10-s + (−0.210 − 0.101i)11-s + (3.24 + 1.00i)12-s + (0.104 − 0.266i)13-s + (−0.721 + 0.669i)14-s + (−2.59 − 1.76i)15-s + (0.616 − 0.772i)16-s + (−0.656 − 0.0989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.674 - 0.738i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.674 - 0.738i)\)
\(L(2)\)  \(\approx\)  \(0.306589 + 0.695014i\)
\(L(\frac12)\)  \(\approx\)  \(0.306589 + 0.695014i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-184. + 213. i)T \)
good2 \( 1 + (1.08 + 4.77i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (6.48 + 6.01i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (-20.4 + 3.07i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (5.26 + 9.12i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (7.67 + 3.69i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (-4.91 + 12.5i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (46.0 + 6.93i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (-1.12 + 15.0i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-125. + 85.2i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (36.7 - 34.1i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (114. + 35.2i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (77.8 - 134. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (56.9 + 249. i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (42.2 - 20.3i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (-27.1 - 69.2i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (531. - 666. i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (-878. + 271. i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (-2.74 + 36.6i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (-670. - 457. i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (-235. + 599. i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-254. - 440. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (37.4 + 34.7i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (-652. - 605. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (709. + 341. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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

−13.70290839954813954883544806617, −13.12507494295808277174233192571, −12.42989702826198105942289310077, −11.00152692737104153407276937634, −10.32896107569767047355940694098, −8.995661363303345524786327423528, −6.71229609190411438034549096338, −5.25996406226175836335247746541, −2.20422453433747175016991553074, −0.843970427865145928991548858245, 5.05091426795508135376842071657, 5.82681572923730607438653567830, 6.67733630717904803789231172337, 9.159838570423813430864912616183, 9.704490022558182482064706324555, 10.94513014911417207263239277317, 13.08708208309529932138738296709, 14.40800054177202433187480018892, 15.39570837168968928708208350446, 16.31555445706504831536521026692

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