Properties

Degree 2
Conductor 43
Sign $-0.885 + 0.465i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.8i·2-s − 128. i·3-s − 311.·4-s − 1.18e3i·5-s + 3.06e3·6-s + 1.04e3i·7-s − 1.32e3i·8-s − 9.98e3·9-s + 2.82e4·10-s − 7.72e3·11-s + 4.00e4i·12-s − 3.27e4·13-s − 2.49e4·14-s − 1.52e5·15-s − 4.82e4·16-s + 7.48e4·17-s + ⋯
L(s)  = 1  + 1.48i·2-s − 1.58i·3-s − 1.21·4-s − 1.89i·5-s + 2.36·6-s + 0.435i·7-s − 0.322i·8-s − 1.52·9-s + 2.82·10-s − 0.527·11-s + 1.93i·12-s − 1.14·13-s − 0.648·14-s − 3.01·15-s − 0.736·16-s + 0.895·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.885 + 0.465i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.885 + 0.465i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.119092 - 0.482825i\)
\(L(\frac12)\)  \(\approx\)  \(0.119092 - 0.482825i\)
\(L(5)\)   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 \( 1 + (3.02e6 - 1.58e6i)T \)
good2 \( 1 - 23.8iT - 256T^{2} \)
3 \( 1 + 128. iT - 6.56e3T^{2} \)
5 \( 1 + 1.18e3iT - 3.90e5T^{2} \)
7 \( 1 - 1.04e3iT - 5.76e6T^{2} \)
11 \( 1 + 7.72e3T + 2.14e8T^{2} \)
13 \( 1 + 3.27e4T + 8.15e8T^{2} \)
17 \( 1 - 7.48e4T + 6.97e9T^{2} \)
19 \( 1 - 2.21e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.72e5T + 7.83e10T^{2} \)
29 \( 1 + 1.10e4iT - 5.00e11T^{2} \)
31 \( 1 + 6.69e5T + 8.52e11T^{2} \)
37 \( 1 + 3.02e6iT - 3.51e12T^{2} \)
41 \( 1 + 8.74e5T + 7.98e12T^{2} \)
47 \( 1 + 2.23e6T + 2.38e13T^{2} \)
53 \( 1 + 1.95e6T + 6.22e13T^{2} \)
59 \( 1 + 2.07e7T + 1.46e14T^{2} \)
61 \( 1 + 1.01e7iT - 1.91e14T^{2} \)
67 \( 1 + 2.69e7T + 4.06e14T^{2} \)
71 \( 1 + 1.28e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.98e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.68e7T + 1.51e15T^{2} \)
83 \( 1 - 2.60e7T + 2.25e15T^{2} \)
89 \( 1 - 2.43e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.57e6T + 7.83e15T^{2} \)
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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

−13.70424698891718582702027454267, −12.62128052761315049469285976911, −12.16688972169995278469652128099, −9.234809315555665851102137097110, −8.088369493876860021586825446186, −7.54255134675384126151646058499, −5.88970737183415357030984856396, −5.08877956657316333211449535005, −1.72088461779461490496802889897, −0.18068095920229381643747090808, 2.71387400771450752016386112094, 3.35458231103426293351478771001, 4.81596554112247689506519198076, 7.10102596004028386436246097128, 9.465378058975866122407773745201, 10.28889349477026895033525033573, 10.79267594306182512360370077953, 11.68682599723590766077586909053, 13.55838852851733852416365392863, 14.78646824946010624867876507329

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