Properties

Degree 2
Conductor 43
Sign $-0.0558 + 0.998i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.3i·2-s + 12.8i·3-s − 171.·4-s + 213. i·5-s − 197.·6-s + 274. i·7-s − 1.65e3i·8-s + 563.·9-s − 3.28e3·10-s + 1.53e3·11-s − 2.20e3i·12-s − 2.26e3·13-s − 4.21e3·14-s − 2.74e3·15-s + 1.44e4·16-s + 8.92e3·17-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.475i·3-s − 2.68·4-s + 1.71i·5-s − 0.913·6-s + 0.800i·7-s − 3.23i·8-s + 0.773·9-s − 3.28·10-s + 1.15·11-s − 1.27i·12-s − 1.02·13-s − 1.53·14-s − 0.814·15-s + 3.52·16-s + 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0558 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0558 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0558 + 0.998i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.0558 + 0.998i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.978160 - 1.03441i\)
\(L(\frac12)\)  \(\approx\)  \(0.978160 - 1.03441i\)
\(L(4)\)   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 \( 1 + (-4.44e3 + 7.93e4i)T \)
good2 \( 1 - 15.3iT - 64T^{2} \)
3 \( 1 - 12.8iT - 729T^{2} \)
5 \( 1 - 213. iT - 1.56e4T^{2} \)
7 \( 1 - 274. iT - 1.17e5T^{2} \)
11 \( 1 - 1.53e3T + 1.77e6T^{2} \)
13 \( 1 + 2.26e3T + 4.82e6T^{2} \)
17 \( 1 - 8.92e3T + 2.41e7T^{2} \)
19 \( 1 + 851. iT - 4.70e7T^{2} \)
23 \( 1 - 1.90e3T + 1.48e8T^{2} \)
29 \( 1 - 8.47e3iT - 5.94e8T^{2} \)
31 \( 1 + 1.47e4T + 8.87e8T^{2} \)
37 \( 1 + 3.96e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.20e4T + 4.75e9T^{2} \)
47 \( 1 - 8.41e4T + 1.07e10T^{2} \)
53 \( 1 + 1.39e5T + 2.21e10T^{2} \)
59 \( 1 - 1.06e5T + 4.21e10T^{2} \)
61 \( 1 - 8.95e4iT - 5.15e10T^{2} \)
67 \( 1 - 2.21e5T + 9.04e10T^{2} \)
71 \( 1 - 2.38e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.59e5iT - 1.51e11T^{2} \)
79 \( 1 - 2.32e5T + 2.43e11T^{2} \)
83 \( 1 - 3.73e5T + 3.26e11T^{2} \)
89 \( 1 - 2.77e5iT - 4.96e11T^{2} \)
97 \( 1 - 8.10e5T + 8.32e11T^{2} \)
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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

−15.27905245330810107843075336350, −14.74741362532561350431688689086, −14.10392089643433262621064581972, −12.26699588313797414431276160579, −10.18775089558408636583882801632, −9.277022081566445977426169546586, −7.52844606810338860259843496485, −6.74736544573938957234495330902, −5.44775576149231790536646805980, −3.68261610040712349260206047620, 0.799266627297453240647830662630, 1.50692198046181224653064858086, 3.88219980864590462226814996698, 4.95991092008331375032491542953, 7.920601300328048995852585336148, 9.357648332881459855993676887165, 10.06970629023699835308725677049, 11.88005196729724703718074102928, 12.39120727070077336001053904843, 13.23670580176268728469670188421

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