Properties

Degree 2
Conductor 43
Sign $0.911 + 0.411i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 59.1i·2-s − 419. i·3-s − 2.46e3·4-s + 832. i·5-s + 2.48e4·6-s + 3.12e4i·7-s − 8.54e4i·8-s − 1.17e5·9-s − 4.92e4·10-s − 3.06e4·11-s + 1.03e6i·12-s − 1.10e4·13-s − 1.84e6·14-s + 3.49e5·15-s + 2.51e6·16-s + 1.14e6·17-s + ⋯
L(s)  = 1  + 1.84i·2-s − 1.72i·3-s − 2.41·4-s + 0.266i·5-s + 3.19·6-s + 1.85i·7-s − 2.60i·8-s − 1.98·9-s − 0.492·10-s − 0.190·11-s + 4.16i·12-s − 0.0296·13-s − 3.43·14-s + 0.460·15-s + 2.40·16-s + 0.807·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.911 + 0.411i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.911 + 0.411i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.836463 - 0.179839i\)
\(L(\frac12)\)  \(\approx\)  \(0.836463 - 0.179839i\)
\(L(6)\)   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 \( 1 + (1.34e8 + 6.04e7i)T \)
good2 \( 1 - 59.1iT - 1.02e3T^{2} \)
3 \( 1 + 419. iT - 5.90e4T^{2} \)
5 \( 1 - 832. iT - 9.76e6T^{2} \)
7 \( 1 - 3.12e4iT - 2.82e8T^{2} \)
11 \( 1 + 3.06e4T + 2.59e10T^{2} \)
13 \( 1 + 1.10e4T + 1.37e11T^{2} \)
17 \( 1 - 1.14e6T + 2.01e12T^{2} \)
19 \( 1 + 4.19e6iT - 6.13e12T^{2} \)
23 \( 1 + 6.78e6T + 4.14e13T^{2} \)
29 \( 1 + 2.48e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.61e6T + 8.19e14T^{2} \)
37 \( 1 + 1.01e8iT - 4.80e15T^{2} \)
41 \( 1 + 3.57e7T + 1.34e16T^{2} \)
47 \( 1 - 2.71e8T + 5.25e16T^{2} \)
53 \( 1 - 4.96e8T + 1.74e17T^{2} \)
59 \( 1 + 6.47e8T + 5.11e17T^{2} \)
61 \( 1 - 5.21e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.81e9T + 1.82e18T^{2} \)
71 \( 1 - 6.55e8iT - 3.25e18T^{2} \)
73 \( 1 + 2.34e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.31e9T + 9.46e18T^{2} \)
83 \( 1 + 3.83e9T + 1.55e19T^{2} \)
89 \( 1 + 7.01e9iT - 3.11e19T^{2} \)
97 \( 1 + 5.78e9T + 7.37e19T^{2} \)
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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

−13.85918188311077414130315700953, −12.84883979356768618951930659030, −11.89022461506434074353523715526, −9.107688871060666526146993632668, −8.244941847225796837787230926969, −7.20575777397408137150626801210, −6.17668350450977313382798318379, −5.38138189517788733601106645402, −2.43674246018632311550365459763, −0.32192040582889467752120179020, 1.16827822551596270692899609020, 3.37389845496612599878138289051, 3.99866311077855998168604155421, 5.04463345047622981428237285630, 8.296113672069417786185533043233, 9.828307823277481105293058875664, 10.23735435793574599524772330936, 10.98084488461889003121697794331, 12.31077602232552305286580975367, 13.77714657038898816610817356179

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