Properties

Degree 2
Conductor 43
Sign $-0.661 - 0.750i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.60 − 4.52i)2-s + (−0.0525 − 0.00791i)3-s + (−0.333 + 1.46i)4-s + (−19.3 − 13.1i)5-s + (0.153 + 0.266i)6-s + (53.2 − 92.1i)7-s + (−159. + 76.5i)8-s + (−232. − 71.6i)9-s + (10.1 + 134. i)10-s + (72.6 + 318. i)11-s + (0.0291 − 0.0741i)12-s + (−28.9 + 386. i)13-s + (−609. + 91.8i)14-s + (0.909 + 0.844i)15-s + (963. + 464. i)16-s + (−1.25e3 + 854. i)17-s + ⋯
L(s)  = 1  + (−0.637 − 0.799i)2-s + (−0.00336 − 0.000507i)3-s + (−0.0104 + 0.0457i)4-s + (−0.345 − 0.235i)5-s + (0.00174 + 0.00301i)6-s + (0.410 − 0.711i)7-s + (−0.878 + 0.423i)8-s + (−0.955 − 0.294i)9-s + (0.0319 + 0.426i)10-s + (0.181 + 0.793i)11-s + (5.83e−5 − 0.000148i)12-s + (−0.0474 + 0.633i)13-s + (−0.830 + 0.125i)14-s + (0.00104 + 0.000968i)15-s + (0.941 + 0.453i)16-s + (−1.05 + 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.661 - 0.750i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.661 - 0.750i)\)
\(L(3)\)  \(\approx\)  \(0.110732 + 0.245294i\)
\(L(\frac12)\)  \(\approx\)  \(0.110732 + 0.245294i\)
\(L(\frac{7}{2})\)   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 \( 1 + (8.51e3 - 8.62e3i)T \)
good2 \( 1 + (3.60 + 4.52i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (0.0525 + 0.00791i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (19.3 + 13.1i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-53.2 + 92.1i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-72.6 - 318. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (28.9 - 386. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (1.25e3 - 854. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (1.79e3 - 553. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (-1.87e3 + 1.74e3i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (3.58e3 - 540. i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (-3.63e3 + 9.26e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (5.99e3 + 1.03e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (-3.84e3 - 4.82e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (232. - 1.01e3i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (554. + 7.40e3i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (1.60e4 + 7.70e3i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (1.56e4 + 3.99e4i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-2.57e4 + 7.94e3i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (1.92e3 + 1.78e3i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (1.54e3 - 2.05e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (-2.60e4 + 4.50e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.05e5 + 1.58e4i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (-9.27e3 - 1.39e3i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (1.33e4 + 5.86e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.36348074559327641599024930389, −12.68716935090779268257235007916, −11.48206570514264113319948776262, −10.70238951710801588956085781163, −9.324348694529955240101927164368, −8.235185845654922306533385873129, −6.35992812875592348577619867143, −4.27656392127580601088807151031, −2.06572767946042050715253606243, −0.16921906422307420616959887203, 3.00522222241833389699772169914, 5.47716690325731668623113360800, 6.91443342886322350321897267649, 8.352873595509491957634975803544, 8.947223159331459198942132021270, 10.97609742909176352808022923272, 11.94193503704289568273434115071, 13.51908636527557368205429549986, 15.01331083345898822607845282820, 15.61962890453098561169223997409

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