Properties

Degree 2
Conductor 43
Sign $-0.518 + 0.855i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.58i·2-s − 3.59i·3-s − 2.67·4-s + 7.02i·5-s − 9.28·6-s − 0.845i·7-s − 3.42i·8-s − 3.91·9-s + 18.1·10-s + 14.5·11-s + 9.60i·12-s − 15.5·13-s − 2.18·14-s + 25.2·15-s − 19.5·16-s + 6.95·17-s + ⋯
L(s)  = 1  − 1.29i·2-s − 1.19i·3-s − 0.668·4-s + 1.40i·5-s − 1.54·6-s − 0.120i·7-s − 0.428i·8-s − 0.435·9-s + 1.81·10-s + 1.32·11-s + 0.800i·12-s − 1.19·13-s − 0.156·14-s + 1.68·15-s − 1.22·16-s + 0.409·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.518 + 0.855i$
motivic weight  =  \(2\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1),\ -0.518 + 0.855i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.553630 - 0.982598i\)
\(L(\frac12)\)  \(\approx\)  \(0.553630 - 0.982598i\)
\(L(2)\)   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 \( 1 + (-22.2 + 36.7i)T \)
good2 \( 1 + 2.58iT - 4T^{2} \)
3 \( 1 + 3.59iT - 9T^{2} \)
5 \( 1 - 7.02iT - 25T^{2} \)
7 \( 1 + 0.845iT - 49T^{2} \)
11 \( 1 - 14.5T + 121T^{2} \)
13 \( 1 + 15.5T + 169T^{2} \)
17 \( 1 - 6.95T + 289T^{2} \)
19 \( 1 - 30.8iT - 361T^{2} \)
23 \( 1 - 17.5T + 529T^{2} \)
29 \( 1 - 7.86iT - 841T^{2} \)
31 \( 1 + 57.7T + 961T^{2} \)
37 \( 1 - 32.5iT - 1.36e3T^{2} \)
41 \( 1 + 18.4T + 1.68e3T^{2} \)
47 \( 1 + 25.7T + 2.20e3T^{2} \)
53 \( 1 + 79.9T + 2.80e3T^{2} \)
59 \( 1 - 18.4T + 3.48e3T^{2} \)
61 \( 1 + 76.1iT - 3.72e3T^{2} \)
67 \( 1 - 9.00T + 4.48e3T^{2} \)
71 \( 1 + 51.6iT - 5.04e3T^{2} \)
73 \( 1 + 77.4iT - 5.32e3T^{2} \)
79 \( 1 - 7.04T + 6.24e3T^{2} \)
83 \( 1 - 83.8T + 6.88e3T^{2} \)
89 \( 1 - 91.8iT - 7.92e3T^{2} \)
97 \( 1 + 155.T + 9.40e3T^{2} \)
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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

−14.81269339658434182781609587498, −14.01169589425783747679852636620, −12.56808531445437474164691555065, −11.91660772602779513629996105157, −10.76569511717795422107149299104, −9.645894059609633394482962084027, −7.42022048927744457369328758716, −6.56021242499329527371779198649, −3.45516535162130046971470401505, −1.82521485635335349723662469855, 4.47745039964027042511682861545, 5.34784569095217386448471058573, 7.18485048000757954750595580541, 8.913803150925598381039744520957, 9.394469689944246412383237085738, 11.34965452722083207473226912122, 12.78492960907881880429444205803, 14.41633296556990520018825911495, 15.19910260650032828533276724161, 16.21563503677180432261377881817

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