Properties

Degree 2
Conductor 43
Sign $0.860 - 0.509i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 + 3.25i)2-s + (17.4 + 2.62i)3-s + (3.25 − 14.2i)4-s + (30.9 + 21.1i)5-s + (36.6 + 63.5i)6-s + (16.4 − 28.4i)7-s + (175. − 84.3i)8-s + (64.0 + 19.7i)9-s + (11.6 + 155. i)10-s + (−51.4 − 225. i)11-s + (94.1 − 239. i)12-s + (−69.9 + 932. i)13-s + (135. − 20.4i)14-s + (483. + 449. i)15-s + (307. + 148. i)16-s + (−1.14e3 + 781. i)17-s + ⋯
L(s)  = 1  + (0.459 + 0.576i)2-s + (1.11 + 0.168i)3-s + (0.101 − 0.445i)4-s + (0.554 + 0.377i)5-s + (0.416 + 0.720i)6-s + (0.126 − 0.219i)7-s + (0.967 − 0.465i)8-s + (0.263 + 0.0812i)9-s + (0.0369 + 0.492i)10-s + (−0.128 − 0.561i)11-s + (0.188 − 0.480i)12-s + (−0.114 + 1.53i)13-s + (0.184 − 0.0278i)14-s + (0.555 + 0.515i)15-s + (0.300 + 0.144i)16-s + (−0.961 + 0.655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.860 - 0.509i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.860 - 0.509i)\)
\(L(3)\)  \(\approx\)  \(2.97851 + 0.815990i\)
\(L(\frac12)\)  \(\approx\)  \(2.97851 + 0.815990i\)
\(L(\frac{7}{2})\)   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 \( 1 + (6.88e3 + 9.98e3i)T \)
good2 \( 1 + (-2.59 - 3.25i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (-17.4 - 2.62i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (-30.9 - 21.1i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-16.4 + 28.4i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (51.4 + 225. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (69.9 - 932. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (1.14e3 - 781. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-1.32e3 + 407. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (911. - 845. i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (4.24e3 - 639. i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (-1.43e3 + 3.65e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (591. + 1.02e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (2.71e3 + 3.40e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (-1.25e3 + 5.51e3i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (-1.54e3 - 2.05e4i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (-2.65e4 - 1.27e4i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (-1.06e4 - 2.71e4i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-6.59e3 + 2.03e3i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (2.70e3 + 2.51e3i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (-921. + 1.23e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (3.70e4 - 6.40e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (7.26e4 + 1.09e4i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (-5.25e4 - 7.92e3i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (8.97e3 + 3.93e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.83002450283869544974675898411, −13.91359184359003325300383109007, −13.57024923222235570644793652253, −11.34484064407705597644657383080, −9.966092488315360872044676641654, −8.843172928918798835683622666518, −7.22082124462531401506069635976, −5.92661094861201913472939416027, −4.09013305483480108552207984751, −2.08105081861565587771389004714, 2.07466514938537033771103189978, 3.26835616263612606328153017957, 5.14568812049967634813856148114, 7.50769523807561539119768636189, 8.537515604388425737152908475608, 9.901309536231280092837373550536, 11.48319096497469148728954398034, 12.86551803081109877246704090848, 13.39694663572565480764228366998, 14.54267550472877714553281247555

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