Properties

Degree 2
Conductor 43
Sign $0.106 + 0.994i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25.2·2-s + (−62.6 + 108. i)3-s + 124.·4-s + (1.18e3 − 2.05e3i)5-s + (1.58e3 − 2.73e3i)6-s + (1.70e3 + 2.95e3i)7-s + 9.78e3·8-s + (1.98e3 + 3.44e3i)9-s + (−2.99e4 + 5.18e4i)10-s − 9.33e4·11-s + (−7.78e3 + 1.34e4i)12-s + (−1.29e4 − 2.23e4i)13-s + (−4.29e4 − 7.44e4i)14-s + (1.48e5 + 2.57e5i)15-s − 3.10e5·16-s + (3.12e5 + 5.40e5i)17-s + ⋯
L(s)  = 1  − 1.11·2-s + (−0.446 + 0.773i)3-s + 0.242·4-s + (0.848 − 1.47i)5-s + (0.497 − 0.862i)6-s + (0.268 + 0.464i)7-s + 0.844·8-s + (0.101 + 0.175i)9-s + (−0.946 + 1.63i)10-s − 1.92·11-s + (−0.108 + 0.187i)12-s + (−0.125 − 0.217i)13-s + (−0.299 − 0.517i)14-s + (0.758 + 1.31i)15-s − 1.18·16-s + (0.906 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.106 + 0.994i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.106 + 0.994i)\)
\(L(5)\)  \(\approx\)  \(0.425842 - 0.382755i\)
\(L(\frac12)\)  \(\approx\)  \(0.425842 - 0.382755i\)
\(L(\frac{11}{2})\)   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 \( 1 + (1.68e7 + 1.48e7i)T \)
good2 \( 1 + 25.2T + 512T^{2} \)
3 \( 1 + (62.6 - 108. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-1.18e3 + 2.05e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-1.70e3 - 2.95e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + 9.33e4T + 2.35e9T^{2} \)
13 \( 1 + (1.29e4 + 2.23e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (-3.12e5 - 5.40e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-1.20e5 + 2.07e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-1.04e6 + 1.80e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-9.75e5 - 1.69e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (-1.47e5 + 2.55e5i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-1.49e6 + 2.59e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 1.70e7T + 3.27e14T^{2} \)
47 \( 1 - 2.92e7T + 1.11e15T^{2} \)
53 \( 1 + (-4.36e7 + 7.55e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + 8.12e7T + 8.66e15T^{2} \)
61 \( 1 + (1.93e7 + 3.35e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-1.31e8 + 2.28e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (6.08e7 + 1.05e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (1.08e8 + 1.88e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (2.06e8 + 3.58e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (7.11e7 - 1.23e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (2.01e7 - 3.48e7i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.95e8T + 7.60e17T^{2} \)
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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

−13.40714905078508752569003926499, −12.60960258433583676046798296611, −10.56783159421906365575823952683, −10.10936482776151997218251571298, −8.797433491306261931926582155244, −8.021135458316072976371772618781, −5.41614782653872681808859344566, −4.81732363046316989216023269556, −1.88101953874072051029002972718, −0.35257014808331262348936066829, 1.16630931372310240942173606894, 2.71858080725541966447795692749, 5.46567722735899900564752340580, 7.14849247187738292504217448920, 7.60023314039797879478731437119, 9.680061678538985367396837324947, 10.37612128574965049941283334470, 11.46745111590808349731553623880, 13.30541192887660102232389025390, 13.97086482534281097506341935457

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