Properties

Degree 2
Conductor 43
Sign $-0.574 - 0.818i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 57.1i·2-s − 18.6i·3-s − 2.23e3·4-s − 2.20e3i·5-s + 1.06e3·6-s − 1.81e3i·7-s − 6.92e4i·8-s + 5.87e4·9-s + 1.25e5·10-s + 8.48e4·11-s + 4.17e4i·12-s − 1.96e5·13-s + 1.03e5·14-s − 4.10e4·15-s + 1.66e6·16-s + 5.62e5·17-s + ⋯
L(s)  = 1  + 1.78i·2-s − 0.0767i·3-s − 2.18·4-s − 0.704i·5-s + 0.136·6-s − 0.108i·7-s − 2.11i·8-s + 0.994·9-s + 1.25·10-s + 0.527·11-s + 0.167i·12-s − 0.529·13-s + 0.193·14-s − 0.0540·15-s + 1.58·16-s + 0.395·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.574 - 0.818i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.574 - 0.818i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.831844 + 1.60112i\)
\(L(\frac12)\)  \(\approx\)  \(0.831844 + 1.60112i\)
\(L(6)\)   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 \( 1 + (-8.45e7 - 1.20e8i)T \)
good2 \( 1 - 57.1iT - 1.02e3T^{2} \)
3 \( 1 + 18.6iT - 5.90e4T^{2} \)
5 \( 1 + 2.20e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.81e3iT - 2.82e8T^{2} \)
11 \( 1 - 8.48e4T + 2.59e10T^{2} \)
13 \( 1 + 1.96e5T + 1.37e11T^{2} \)
17 \( 1 - 5.62e5T + 2.01e12T^{2} \)
19 \( 1 - 2.47e6iT - 6.13e12T^{2} \)
23 \( 1 - 4.25e6T + 4.14e13T^{2} \)
29 \( 1 + 7.15e3iT - 4.20e14T^{2} \)
31 \( 1 - 9.25e6T + 8.19e14T^{2} \)
37 \( 1 - 2.64e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.16e8T + 1.34e16T^{2} \)
47 \( 1 - 3.14e7T + 5.25e16T^{2} \)
53 \( 1 - 2.99e8T + 1.74e17T^{2} \)
59 \( 1 - 6.98e6T + 5.11e17T^{2} \)
61 \( 1 - 1.07e9iT - 7.13e17T^{2} \)
67 \( 1 - 1.30e9T + 1.82e18T^{2} \)
71 \( 1 - 1.39e9iT - 3.25e18T^{2} \)
73 \( 1 + 7.73e8iT - 4.29e18T^{2} \)
79 \( 1 + 2.98e8T + 9.46e18T^{2} \)
83 \( 1 - 3.52e9T + 1.55e19T^{2} \)
89 \( 1 - 8.07e9iT - 3.11e19T^{2} \)
97 \( 1 + 7.54e9T + 7.37e19T^{2} \)
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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

−14.45469091629507871036956049648, −13.30668820364947235507615510799, −12.29075470998073675193977450341, −9.985493806169127579563382586479, −8.858000891300140611521213068469, −7.69041012199834427076925963833, −6.66176911704819389384163788389, −5.26654786895526743958189439645, −4.16560349468180984601118606912, −1.02630456561437440512285416711, 0.810480869031902552764870608221, 2.24137700929785494338427144649, 3.48783003067293310711487824074, 4.79256555740882594312634463128, 7.04472343252124655346790266707, 9.006779972825428091439186952849, 10.04452658865412375406945534338, 10.91921611305114033181210734231, 12.01498993987638435326431327780, 12.98142955218668445343653316147

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