Properties

Degree 2
Conductor 43
Sign $-0.144 + 0.989i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.0i·2-s + 78.5i·3-s − 273.·4-s + 548. i·5-s − 1.80e3·6-s + 2.21e3i·7-s − 399. i·8-s + 395.·9-s − 1.26e4·10-s − 2.23e4·11-s − 2.14e4i·12-s + 2.57e4·13-s − 5.09e4·14-s − 4.30e4·15-s − 6.07e4·16-s + 9.48e4·17-s + ⋯
L(s)  = 1  + 1.43i·2-s + 0.969i·3-s − 1.06·4-s + 0.877i·5-s − 1.39·6-s + 0.922i·7-s − 0.0974i·8-s + 0.0602·9-s − 1.26·10-s − 1.52·11-s − 1.03i·12-s + 0.900·13-s − 1.32·14-s − 0.850·15-s − 0.927·16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.144 + 0.989i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.144 + 0.989i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.08684 - 1.25695i\)
\(L(\frac12)\)  \(\approx\)  \(1.08684 - 1.25695i\)
\(L(5)\)   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 \( 1 + (4.93e5 - 3.38e6i)T \)
good2 \( 1 - 23.0iT - 256T^{2} \)
3 \( 1 - 78.5iT - 6.56e3T^{2} \)
5 \( 1 - 548. iT - 3.90e5T^{2} \)
7 \( 1 - 2.21e3iT - 5.76e6T^{2} \)
11 \( 1 + 2.23e4T + 2.14e8T^{2} \)
13 \( 1 - 2.57e4T + 8.15e8T^{2} \)
17 \( 1 - 9.48e4T + 6.97e9T^{2} \)
19 \( 1 - 1.37e4iT - 1.69e10T^{2} \)
23 \( 1 - 4.44e5T + 7.83e10T^{2} \)
29 \( 1 + 8.43e5iT - 5.00e11T^{2} \)
31 \( 1 - 5.57e5T + 8.52e11T^{2} \)
37 \( 1 + 1.34e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.42e6T + 7.98e12T^{2} \)
47 \( 1 + 4.12e6T + 2.38e13T^{2} \)
53 \( 1 - 7.17e5T + 6.22e13T^{2} \)
59 \( 1 + 1.58e7T + 1.46e14T^{2} \)
61 \( 1 - 1.76e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.33e7T + 4.06e14T^{2} \)
71 \( 1 + 1.27e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.05e6iT - 8.06e14T^{2} \)
79 \( 1 + 6.41e7T + 1.51e15T^{2} \)
83 \( 1 + 4.42e7T + 2.25e15T^{2} \)
89 \( 1 + 8.42e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.90e7T + 7.83e15T^{2} \)
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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

−15.37219563881707968698716748740, −14.52713725263599969640957445101, −13.11825660911732987690344773914, −11.18776118497176385692580865555, −10.06805256865888012741461433421, −8.651293583149554387589163788630, −7.43800790926512168161129403994, −5.98106276572318579088839323892, −4.95350877033162755873970097051, −2.96773036168082289632937702054, 0.69651122987050327627663698659, 1.43238974412245003418600575797, 3.18804168596234580275011452836, 4.91144287658711010729491143385, 7.07788056571071795404360880724, 8.428690528119905551447327597230, 10.05908650050203727055350907243, 10.99052225010123362616607354500, 12.40579891931471764266738756568, 13.02821198901518315284058254242

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