Properties

Degree 2
Conductor 43
Sign $-0.998 + 0.0620i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.22 + 6.69i)2-s + (−59.4 + 123. i)3-s + (125. − 156. i)4-s + (267. + 61.0i)5-s − 1.01e3·6-s + 4.43e3i·7-s + (3.31e3 + 755. i)8-s + (−7.62e3 − 9.55e3i)9-s + (453. + 1.98e3i)10-s + (9.75e3 + 1.22e4i)11-s + (1.19e4 + 2.47e4i)12-s + (−6.42e3 + 2.81e4i)13-s + (−2.97e4 + 1.43e4i)14-s + (−2.34e4 + 2.93e4i)15-s + (−5.81e3 − 2.54e4i)16-s + (−2.80e4 − 1.23e5i)17-s + ⋯
L(s)  = 1  + (0.201 + 0.418i)2-s + (−0.734 + 1.52i)3-s + (0.488 − 0.613i)4-s + (0.427 + 0.0976i)5-s − 0.786·6-s + 1.84i·7-s + (0.808 + 0.184i)8-s + (−1.16 − 1.45i)9-s + (0.0453 + 0.198i)10-s + (0.666 + 0.835i)11-s + (0.575 + 1.19i)12-s + (−0.225 + 0.985i)13-s + (−0.774 + 0.372i)14-s + (−0.462 + 0.580i)15-s + (−0.0887 − 0.388i)16-s + (−0.336 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.998 + 0.0620i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.998 + 0.0620i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.0528905 - 1.70207i\)
\(L(\frac12)\)  \(\approx\)  \(0.0528905 - 1.70207i\)
\(L(5)\)   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 \( 1 + (-2.84e6 + 1.90e6i)T \)
good2 \( 1 + (-3.22 - 6.69i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (59.4 - 123. i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-267. - 61.0i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 - 4.43e3iT - 5.76e6T^{2} \)
11 \( 1 + (-9.75e3 - 1.22e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (6.42e3 - 2.81e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (2.80e4 + 1.23e5i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (3.27e4 + 2.61e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-7.97e4 - 1.00e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (1.82e5 + 3.78e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (4.47e5 - 2.15e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 - 3.20e6iT - 3.51e12T^{2} \)
41 \( 1 + (-1.45e6 + 6.98e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (2.10e6 - 2.64e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (8.71e4 + 3.81e5i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (-3.76e6 - 1.64e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-8.30e6 + 1.72e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (1.58e6 - 1.98e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (-5.07e6 - 4.05e6i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-4.41e7 - 1.00e7i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 1.21e7T + 1.51e15T^{2} \)
83 \( 1 + (-1.40e7 - 6.75e6i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (3.66e7 - 7.61e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (-6.71e7 - 8.42e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−15.20584922111058617962511881291, −14.20412233391452325096689408567, −11.92897914333034906692469008899, −11.31962450125008742680741348735, −9.730347294010994382528332197435, −9.261310063446534809187106134237, −6.59472414133705097895551945530, −5.52960317689371001171472429398, −4.64886898781942979587435507264, −2.26856165067421472677748374563, 0.65261388926598661166554242459, 1.76070120764871798145380500607, 3.76681194967174090359285710474, 6.06001297310945083332241129500, 7.13971970815260448817897613236, 8.025592622482737523058503009543, 10.61131034367546271089061876642, 11.23870477137217183015287154674, 12.76534358663523320155101990948, 13.08045037189967574765907891423

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