Properties

Degree 2
Conductor 43
Sign $0.453 + 0.891i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 44.1i·2-s − 389. i·3-s − 927.·4-s + 5.23e3i·5-s − 1.71e4·6-s + 2.18e4i·7-s − 4.27e3i·8-s − 9.24e4·9-s + 2.31e5·10-s + 1.62e4·11-s + 3.60e5i·12-s + 3.26e5·13-s + 9.67e5·14-s + 2.03e6·15-s − 1.13e6·16-s + 2.87e5·17-s + ⋯
L(s)  = 1  − 1.38i·2-s − 1.60i·3-s − 0.905·4-s + 1.67i·5-s − 2.21·6-s + 1.30i·7-s − 0.130i·8-s − 1.56·9-s + 2.31·10-s + 0.100·11-s + 1.45i·12-s + 0.878·13-s + 1.79·14-s + 2.68·15-s − 1.08·16-s + 0.202·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.453 + 0.891i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.453 + 0.891i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(1.51481 - 0.928471i\)
\(L(\frac12)\)  \(\approx\)  \(1.51481 - 0.928471i\)
\(L(6)\)   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 \( 1 + (6.67e7 + 1.30e8i)T \)
good2 \( 1 + 44.1iT - 1.02e3T^{2} \)
3 \( 1 + 389. iT - 5.90e4T^{2} \)
5 \( 1 - 5.23e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.18e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.62e4T + 2.59e10T^{2} \)
13 \( 1 - 3.26e5T + 1.37e11T^{2} \)
17 \( 1 - 2.87e5T + 2.01e12T^{2} \)
19 \( 1 - 1.39e5iT - 6.13e12T^{2} \)
23 \( 1 - 1.90e6T + 4.14e13T^{2} \)
29 \( 1 - 3.83e7iT - 4.20e14T^{2} \)
31 \( 1 - 5.40e7T + 8.19e14T^{2} \)
37 \( 1 - 8.73e7iT - 4.80e15T^{2} \)
41 \( 1 - 8.87e7T + 1.34e16T^{2} \)
47 \( 1 - 1.99e8T + 5.25e16T^{2} \)
53 \( 1 + 3.58e8T + 1.74e17T^{2} \)
59 \( 1 - 9.43e8T + 5.11e17T^{2} \)
61 \( 1 - 1.50e9iT - 7.13e17T^{2} \)
67 \( 1 + 1.23e9T + 1.82e18T^{2} \)
71 \( 1 + 1.85e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.84e9iT - 4.29e18T^{2} \)
79 \( 1 + 8.05e8T + 9.46e18T^{2} \)
83 \( 1 + 6.35e9T + 1.55e19T^{2} \)
89 \( 1 - 3.36e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.34e10T + 7.37e19T^{2} \)
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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

−13.21181485920502800513137858279, −12.09603083892420887519112957465, −11.48568374548139022804209043591, −10.36909510262075156652481699378, −8.653262354523012920053139217189, −7.03787326186116070731715207527, −6.12847907638449826868900934980, −3.13105712873850286055980090951, −2.43759929736859434154833366627, −1.26567612365102684636483348264, 0.64804307152826066463629428214, 4.09919687339472144258736173361, 4.72657107756043359080177018274, 5.97474501469927418737993152082, 7.903653247555090333479551558656, 8.872907490498346369175887011015, 9.933304165700387732532356963332, 11.35150596463974497275532433260, 13.28774051470292506014033567968, 14.24816840604942010437551710449

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