Properties

Degree 2
Conductor 43
Sign $0.284 + 0.958i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 43.4i·2-s + 375. i·3-s − 864.·4-s + 561. i·5-s − 1.63e4·6-s + 2.62e4i·7-s + 6.94e3i·8-s − 8.21e4·9-s − 2.44e4·10-s + 1.76e5·11-s − 3.24e5i·12-s − 4.66e5·13-s − 1.14e6·14-s − 2.11e5·15-s − 1.18e6·16-s + 2.37e5·17-s + ⋯
L(s)  = 1  + 1.35i·2-s + 1.54i·3-s − 0.843·4-s + 0.179i·5-s − 2.09·6-s + 1.56i·7-s + 0.212i·8-s − 1.39·9-s − 0.244·10-s + 1.09·11-s − 1.30i·12-s − 1.25·13-s − 2.12·14-s − 0.278·15-s − 1.13·16-s + 0.167·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.284 + 0.958i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.284 + 0.958i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(1.48258 - 1.10690i\)
\(L(\frac12)\)  \(\approx\)  \(1.48258 - 1.10690i\)
\(L(6)\)   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 \( 1 + (4.17e7 + 1.40e8i)T \)
good2 \( 1 - 43.4iT - 1.02e3T^{2} \)
3 \( 1 - 375. iT - 5.90e4T^{2} \)
5 \( 1 - 561. iT - 9.76e6T^{2} \)
7 \( 1 - 2.62e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.76e5T + 2.59e10T^{2} \)
13 \( 1 + 4.66e5T + 1.37e11T^{2} \)
17 \( 1 - 2.37e5T + 2.01e12T^{2} \)
19 \( 1 + 5.19e5iT - 6.13e12T^{2} \)
23 \( 1 - 1.16e7T + 4.14e13T^{2} \)
29 \( 1 - 2.76e7iT - 4.20e14T^{2} \)
31 \( 1 - 6.10e6T + 8.19e14T^{2} \)
37 \( 1 - 1.86e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.74e8T + 1.34e16T^{2} \)
47 \( 1 - 3.46e8T + 5.25e16T^{2} \)
53 \( 1 - 2.36e8T + 1.74e17T^{2} \)
59 \( 1 + 8.09e8T + 5.11e17T^{2} \)
61 \( 1 + 4.73e8iT - 7.13e17T^{2} \)
67 \( 1 - 2.10e8T + 1.82e18T^{2} \)
71 \( 1 + 1.02e9iT - 3.25e18T^{2} \)
73 \( 1 - 7.03e8iT - 4.29e18T^{2} \)
79 \( 1 + 1.85e9T + 9.46e18T^{2} \)
83 \( 1 - 9.78e8T + 1.55e19T^{2} \)
89 \( 1 + 4.53e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.30e10T + 7.37e19T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.89433122144106521389166915197, −14.48925904085107354754811178430, −12.24225762609867446401058173090, −10.95252162334598420655292800538, −9.256582643640428559280118984222, −8.864857864019723718997593427356, −6.97153161789410289994527126079, −5.53328462678068773785624426487, −4.73676642902527017391079092092, −2.82626408759234350439531274957, 0.69484646391007882738646138004, 1.21141277153675673249217649135, 2.67084101346042255328551575860, 4.27584810885023225003449290208, 6.72976681459970449411015831584, 7.50417690816217661125888028334, 9.364978533287860553203449017567, 10.71258339119490396426818973070, 11.80490808961625487979672890450, 12.67771536327627943937001698354

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