Properties

Degree 2
Conductor 43
Sign $-0.954 + 0.299i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 − 2.24i)2-s + (−24.6 + 51.1i)3-s + (155. − 195. i)4-s + (−234. − 53.6i)5-s + 141.·6-s − 271. i·7-s + (−1.22e3 − 280. i)8-s + (2.07e3 + 2.60e3i)9-s + (133. + 585. i)10-s + (−8.44e3 − 1.05e4i)11-s + (6.15e3 + 1.27e4i)12-s + (−5.83e3 + 2.55e4i)13-s + (−610. + 293. i)14-s + (8.53e3 − 1.07e4i)15-s + (−1.35e4 − 5.92e4i)16-s + (4.85e3 + 2.12e4i)17-s + ⋯
L(s)  = 1  + (−0.0675 − 0.140i)2-s + (−0.304 + 0.631i)3-s + (0.608 − 0.762i)4-s + (−0.375 − 0.0858i)5-s + 0.109·6-s − 0.113i·7-s + (−0.300 − 0.0684i)8-s + (0.316 + 0.397i)9-s + (0.0133 + 0.0585i)10-s + (−0.576 − 0.723i)11-s + (0.296 + 0.616i)12-s + (−0.204 + 0.894i)13-s + (−0.0158 + 0.00764i)14-s + (0.168 − 0.211i)15-s + (−0.206 − 0.904i)16-s + (0.0581 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.954 + 0.299i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.954 + 0.299i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.0565926 - 0.369219i\)
\(L(\frac12)\)  \(\approx\)  \(0.0565926 - 0.369219i\)
\(L(5)\)   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 \( 1 + (-2.30e6 + 2.52e6i)T \)
good2 \( 1 + (1.08 + 2.24i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (24.6 - 51.1i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (234. + 53.6i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 271. iT - 5.76e6T^{2} \)
11 \( 1 + (8.44e3 + 1.05e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (5.83e3 - 2.55e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (-4.85e3 - 2.12e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (1.39e5 + 1.11e5i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (2.57e5 + 3.23e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-2.66e5 - 5.52e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (1.48e6 - 7.17e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 2.79e6iT - 3.51e12T^{2} \)
41 \( 1 + (2.88e6 - 1.38e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-3.23e6 + 4.05e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-2.29e6 - 1.00e7i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (-3.62e6 - 1.58e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (8.22e5 - 1.70e6i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (-6.93e6 + 8.69e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (3.08e6 + 2.45e6i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (4.32e7 + 9.86e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 - 6.29e7T + 1.51e15T^{2} \)
83 \( 1 + (3.31e7 + 1.59e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (3.76e7 - 7.81e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (2.48e7 + 3.11e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−13.84590159075899898565381513113, −12.24007501387615706922745743217, −10.89615927100255719069192181071, −10.43231523613889074289457887608, −8.877191595086064582370117816810, −7.12774490013465451242322199893, −5.65184305677055849588351271376, −4.24487951027500211770254654493, −2.13517593234758012400023884323, −0.13651143680054979740092210446, 2.01893646249480641687801108852, 3.76526648784944297526694696942, 5.93741608032913709024078223020, 7.31339979585365427795472024721, 8.024516487276438673396026229532, 9.953786521565541958910318744611, 11.49677351926477151376697622106, 12.37836610177818681818503010187, 13.15015909409267123328130755696, 15.09014305832642018340260403033

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