Properties

Degree 2
Conductor 43
Sign $0.814 + 0.580i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.4i·2-s + (3.88 + 2.24i)3-s − 91.6·4-s + (−151. − 87.2i)5-s + (−27.9 + 48.4i)6-s + (378. − 218. i)7-s − 344. i·8-s + (−354. − 613. i)9-s + (1.08e3 − 1.88e3i)10-s − 420.·11-s + (−355. − 205. i)12-s + (−379. − 657. i)13-s + (2.72e3 + 4.72e3i)14-s + (−391. − 677. i)15-s − 1.56e3·16-s + (−92.2 − 159. i)17-s + ⋯
L(s)  = 1  + 1.55i·2-s + (0.143 + 0.0830i)3-s − 1.43·4-s + (−1.20 − 0.697i)5-s + (−0.129 + 0.224i)6-s + (1.10 − 0.637i)7-s − 0.673i·8-s + (−0.486 − 0.842i)9-s + (1.08 − 1.88i)10-s − 0.315·11-s + (−0.205 − 0.118i)12-s + (−0.172 − 0.299i)13-s + (0.993 + 1.72i)14-s + (−0.115 − 0.200i)15-s − 0.382·16-s + (−0.0187 − 0.0325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.814 + 0.580i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.814 + 0.580i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.713119 - 0.228332i\)
\(L(\frac12)\)  \(\approx\)  \(0.713119 - 0.228332i\)
\(L(4)\)   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 \( 1 + (-4.04e4 + 6.84e4i)T \)
good2 \( 1 - 12.4iT - 64T^{2} \)
3 \( 1 + (-3.88 - 2.24i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (151. + 87.2i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-378. + 218. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 420.T + 1.77e6T^{2} \)
13 \( 1 + (379. + 657. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (92.2 + 159. i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (703. + 405. i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-6.48e3 + 1.12e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (3.08e4 - 1.78e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-1.35e4 + 2.35e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (6.04e4 + 3.48e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 1.11e5T + 4.75e9T^{2} \)
47 \( 1 - 1.35e5T + 1.07e10T^{2} \)
53 \( 1 + (7.54e4 - 1.30e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 6.12e4T + 4.21e10T^{2} \)
61 \( 1 + (-7.66e4 + 4.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.17e5 + 2.03e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (1.05e5 - 6.11e4i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (2.97e5 - 1.71e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-4.35e5 - 7.54e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (4.20e5 - 7.28e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-6.72e5 - 3.88e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.04e6T + 8.32e11T^{2} \)
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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

−14.94836966846352153430745636645, −13.93332160640399669001517488589, −12.35220248494302422368140477564, −11.05979586931417576874583691890, −8.875008568955742388110328331242, −8.086748853333535054758348481057, −7.11933600038738755175679737337, −5.29537491192317756544825087627, −4.11488914555956751834923932646, −0.34601058867296315927194144876, 1.94430449264990361416433182178, 3.30985225874900385277055951869, 4.89385331660602357460344728454, 7.55970276795986301056811451752, 8.722224197715110606207217268550, 10.48638425280482158638430075359, 11.43083876701748289321348506840, 11.84879022088934532614713042681, 13.39103533565821154902241716919, 14.62759928137233350763244040338

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