Properties

Degree 2
Conductor 43
Sign $-0.871 - 0.490i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.90i·2-s + (12.0 + 6.96i)3-s − 34.1·4-s + (−41.4 − 23.9i)5-s + (69.0 − 119. i)6-s + (−415. + 240. i)7-s − 295. i·8-s + (−267. − 463. i)9-s + (−237. + 410. i)10-s − 1.05e3·11-s + (−411. − 237. i)12-s + (−1.56e3 − 2.71e3i)13-s + (2.37e3 + 4.11e3i)14-s + (−333. − 578. i)15-s − 5.11e3·16-s + (1.14e3 + 1.98e3i)17-s + ⋯
L(s)  = 1  − 1.23i·2-s + (0.447 + 0.258i)3-s − 0.533·4-s + (−0.331 − 0.191i)5-s + (0.319 − 0.553i)6-s + (−1.21 + 0.699i)7-s − 0.578i·8-s + (−0.366 − 0.635i)9-s + (−0.237 + 0.410i)10-s − 0.789·11-s + (−0.238 − 0.137i)12-s + (−0.713 − 1.23i)13-s + (0.866 + 1.50i)14-s + (−0.0988 − 0.171i)15-s − 1.24·16-s + (0.233 + 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.871 - 0.490i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.871 - 0.490i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.195682 + 0.746523i\)
\(L(\frac12)\)  \(\approx\)  \(0.195682 + 0.746523i\)
\(L(4)\)   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 \( 1 + (3.28e4 - 7.23e4i)T \)
good2 \( 1 + 9.90iT - 64T^{2} \)
3 \( 1 + (-12.0 - 6.96i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (41.4 + 23.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (415. - 240. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 1.05e3T + 1.77e6T^{2} \)
13 \( 1 + (1.56e3 + 2.71e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (-1.14e3 - 1.98e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.00e4 - 5.78e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (8.69e3 - 1.50e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-2.38e4 + 1.37e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-2.36e4 + 4.10e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (6.20e4 + 3.58e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 2.36e4T + 4.75e9T^{2} \)
47 \( 1 - 1.54e5T + 1.07e10T^{2} \)
53 \( 1 + (-5.03e4 + 8.72e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 3.80e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.11e5 + 1.21e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.28e5 + 2.22e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (2.40e5 - 1.38e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-8.21e4 + 4.74e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (8.09e4 + 1.40e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (9.80e4 - 1.69e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-2.27e5 - 1.31e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 2.79e4T + 8.32e11T^{2} \)
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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

−13.66252040619410873722001052636, −12.40866875653527891641527723257, −11.93112284061729396437297344305, −10.10270438755386412678651673315, −9.612685377906001150834208350792, −7.940509858808969220468542405008, −5.85459508709312554230876154667, −3.56518109918586855051592302118, −2.69543689036022112630275874037, −0.32020147128093427364800200643, 2.82633843532445578733012643538, 5.02375759503278817994216704796, 6.81646146856361576927537089196, 7.42591054240807272404323394954, 8.812653072034441326940924459317, 10.34982435345983316546932600507, 11.97039589290537912753863063080, 13.71735661774418082965863608532, 14.06127736015232785103895345641, 15.65956688400533653562113063813

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