Properties

Degree 2
Conductor 43
Sign $-0.617 + 0.786i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.02 − 1.37i)2-s + (4.35 − 14.1i)3-s + (20.0 + 9.63i)4-s + (39.0 − 15.3i)5-s + (−45.6 + 79.0i)6-s + (34.4 − 19.8i)7-s + (−29.9 − 23.8i)8-s + (−113. − 77.1i)9-s + (−256. + 38.6i)10-s + (81.4 − 39.2i)11-s + (222. − 240. i)12-s + (−57.9 − 8.72i)13-s + (−234. + 72.4i)14-s + (−46.2 − 617. i)15-s + (−73.7 − 92.5i)16-s + (−195. + 497. i)17-s + ⋯
L(s)  = 1  + (−1.50 − 0.343i)2-s + (0.483 − 1.56i)3-s + (1.25 + 0.602i)4-s + (1.56 − 0.612i)5-s + (−1.26 + 2.19i)6-s + (0.702 − 0.405i)7-s + (−0.468 − 0.373i)8-s + (−1.39 − 0.952i)9-s + (−2.56 + 0.386i)10-s + (0.672 − 0.324i)11-s + (1.54 − 1.66i)12-s + (−0.342 − 0.0516i)13-s + (−1.19 + 0.369i)14-s + (−0.205 − 2.74i)15-s + (−0.288 − 0.361i)16-s + (−0.676 + 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.617 + 0.786i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.617 + 0.786i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.485664 - 0.998730i\)
\(L(\frac12)\)  \(\approx\)  \(0.485664 - 0.998730i\)
\(L(3)\)   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 \( 1 + (790. - 1.67e3i)T \)
good2 \( 1 + (6.02 + 1.37i)T + (14.4 + 6.94i)T^{2} \)
3 \( 1 + (-4.35 + 14.1i)T + (-66.9 - 45.6i)T^{2} \)
5 \( 1 + (-39.0 + 15.3i)T + (458. - 425. i)T^{2} \)
7 \( 1 + (-34.4 + 19.8i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-81.4 + 39.2i)T + (9.12e3 - 1.14e4i)T^{2} \)
13 \( 1 + (57.9 + 8.72i)T + (2.72e4 + 8.41e3i)T^{2} \)
17 \( 1 + (195. - 497. i)T + (-6.12e4 - 5.68e4i)T^{2} \)
19 \( 1 + (-141. - 208. i)T + (-4.76e4 + 1.21e5i)T^{2} \)
23 \( 1 + (43.4 - 579. i)T + (-2.76e5 - 4.17e4i)T^{2} \)
29 \( 1 + (-129. - 419. i)T + (-5.84e5 + 3.98e5i)T^{2} \)
31 \( 1 + (1.13e3 + 1.05e3i)T + (6.90e4 + 9.20e5i)T^{2} \)
37 \( 1 + (115. + 66.5i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-266. + 1.16e3i)T + (-2.54e6 - 1.22e6i)T^{2} \)
47 \( 1 + (-3.04e3 - 1.46e3i)T + (3.04e6 + 3.81e6i)T^{2} \)
53 \( 1 + (3.33e3 - 502. i)T + (7.53e6 - 2.32e6i)T^{2} \)
59 \( 1 + (-1.25e3 - 1.57e3i)T + (-2.69e6 + 1.18e7i)T^{2} \)
61 \( 1 + (275. + 296. i)T + (-1.03e6 + 1.38e7i)T^{2} \)
67 \( 1 + (-685. + 467. i)T + (7.36e6 - 1.87e7i)T^{2} \)
71 \( 1 + (-473. + 35.4i)T + (2.51e7 - 3.78e6i)T^{2} \)
73 \( 1 + (-979. + 6.50e3i)T + (-2.71e7 - 8.37e6i)T^{2} \)
79 \( 1 + (-2.41e3 - 4.18e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.51e3 + 468. i)T + (3.92e7 + 2.67e7i)T^{2} \)
89 \( 1 + (-2.33e3 + 7.55e3i)T + (-5.18e7 - 3.53e7i)T^{2} \)
97 \( 1 + (3.96e3 - 1.90e3i)T + (5.51e7 - 6.92e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.38256928485743343207017338590, −13.52401343134998394318004416405, −12.45870622336540646577677177112, −10.96291879999850057501793250383, −9.513633150965870680965600440888, −8.579182720858584156310782368181, −7.53914823673825706904444630562, −6.06206387042262860591406534520, −1.90837109930236658517262093005, −1.31567027989520850789539660937, 2.35064788646462213120761600009, 5.01596805066164599095140179143, 6.87771813403161048304085725706, 8.832864627176928414356137220517, 9.417461967451533698692431158443, 10.20886809523581084716113831523, 11.21353880575866602748761400898, 13.92178184362524689612015405345, 14.73447687200422365783770719892, 15.82442209706880129307933301553

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