Properties

Degree 2
Conductor 43
Sign $0.755 + 0.655i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.51i·2-s + (20.0 − 11.5i)3-s + 61.6·4-s + (211. − 122. i)5-s + (17.5 + 30.4i)6-s + (−447. − 258. i)7-s + 190. i·8-s + (−96.7 + 167. i)9-s + (185. + 320. i)10-s − 593.·11-s + (1.23e3 − 713. i)12-s + (651. − 1.12e3i)13-s + (391. − 678. i)14-s + (2.82e3 − 4.89e3i)15-s + 3.65e3·16-s + (−1.53e3 + 2.66e3i)17-s + ⋯
L(s)  = 1  + 0.189i·2-s + (0.742 − 0.428i)3-s + 0.964·4-s + (1.69 − 0.976i)5-s + (0.0812 + 0.140i)6-s + (−1.30 − 0.753i)7-s + 0.372i·8-s + (−0.132 + 0.229i)9-s + (0.185 + 0.320i)10-s − 0.446·11-s + (0.715 − 0.413i)12-s + (0.296 − 0.513i)13-s + (0.142 − 0.247i)14-s + (0.836 − 1.44i)15-s + 0.893·16-s + (−0.313 + 0.542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.755 + 0.655i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.755 + 0.655i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(2.75033 - 1.02690i\)
\(L(\frac12)\)  \(\approx\)  \(2.75033 - 1.02690i\)
\(L(4)\)   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 \( 1 + (5.70e4 - 5.53e4i)T \)
good2 \( 1 - 1.51iT - 64T^{2} \)
3 \( 1 + (-20.0 + 11.5i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-211. + 122. i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (447. + 258. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 593.T + 1.77e6T^{2} \)
13 \( 1 + (-651. + 1.12e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (1.53e3 - 2.66e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (3.28e3 - 1.89e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-8.01e3 - 1.38e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-4.72e3 - 2.72e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-7.58e3 - 1.31e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-2.86e3 + 1.65e3i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 8.53e4T + 4.75e9T^{2} \)
47 \( 1 + 2.25e4T + 1.07e10T^{2} \)
53 \( 1 + (1.28e5 + 2.23e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + 1.07e5T + 4.21e10T^{2} \)
61 \( 1 + (1.91e4 + 1.10e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-2.30e5 - 3.99e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-3.46e5 - 2.00e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (4.67e5 + 2.70e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.44e5 - 4.22e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (3.60e5 + 6.23e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (6.12e4 - 3.53e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.36e6T + 8.32e11T^{2} \)
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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

−14.32168506054624905570122344024, −13.13385532648073767723935384118, −12.92967949230992232893337559173, −10.64277690752866958026061014111, −9.630828042948186563578286451468, −8.256043443084421118264204779742, −6.71660863341275932579880294678, −5.60094688589794456947289868968, −2.86457895485629219976711394702, −1.49574881477229280027304285094, 2.41493664219357466068805954147, 2.96984270138587307954039601976, 6.05247225785714271864612496712, 6.70501358677451648232101923671, 9.096310712114442605559317064276, 9.867267458586000435850349717728, 10.92196727129664839139812891051, 12.61847169810163271888234645939, 13.77590358395428149835810058818, 14.87667226773208762612724715619

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