Properties

Degree 2
Conductor 43
Sign $0.955 - 0.293i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.19i·2-s − 89.9i·3-s + 217.·4-s − 140. i·5-s + 557.·6-s + 4.21e3i·7-s + 2.93e3i·8-s − 1.52e3·9-s + 872.·10-s − 4.22e3·11-s − 1.95e4i·12-s + 3.53e4·13-s − 2.61e4·14-s − 1.26e4·15-s + 3.75e4·16-s + 2.30e4·17-s + ⋯
L(s)  = 1  + 0.387i·2-s − 1.11i·3-s + 0.849·4-s − 0.225i·5-s + 0.430·6-s + 1.75i·7-s + 0.716i·8-s − 0.232·9-s + 0.0872·10-s − 0.288·11-s − 0.943i·12-s + 1.23·13-s − 0.680·14-s − 0.250·15-s + 0.572·16-s + 0.276·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.955 - 0.293i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.955 - 0.293i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(2.44413 + 0.366732i\)
\(L(\frac12)\)  \(\approx\)  \(2.44413 + 0.366732i\)
\(L(5)\)   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 \( 1 + (-3.26e6 + 1.00e6i)T \)
good2 \( 1 - 6.19iT - 256T^{2} \)
3 \( 1 + 89.9iT - 6.56e3T^{2} \)
5 \( 1 + 140. iT - 3.90e5T^{2} \)
7 \( 1 - 4.21e3iT - 5.76e6T^{2} \)
11 \( 1 + 4.22e3T + 2.14e8T^{2} \)
13 \( 1 - 3.53e4T + 8.15e8T^{2} \)
17 \( 1 - 2.30e4T + 6.97e9T^{2} \)
19 \( 1 + 1.75e3iT - 1.69e10T^{2} \)
23 \( 1 + 1.05e5T + 7.83e10T^{2} \)
29 \( 1 - 8.07e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.74e5T + 8.52e11T^{2} \)
37 \( 1 + 2.09e6iT - 3.51e12T^{2} \)
41 \( 1 - 3.35e6T + 7.98e12T^{2} \)
47 \( 1 + 3.56e6T + 2.38e13T^{2} \)
53 \( 1 + 5.72e6T + 6.22e13T^{2} \)
59 \( 1 + 1.72e7T + 1.46e14T^{2} \)
61 \( 1 - 1.48e7iT - 1.91e14T^{2} \)
67 \( 1 - 1.67e7T + 4.06e14T^{2} \)
71 \( 1 + 1.93e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.18e7iT - 8.06e14T^{2} \)
79 \( 1 + 2.09e7T + 1.51e15T^{2} \)
83 \( 1 - 1.44e6T + 2.25e15T^{2} \)
89 \( 1 - 5.12e7iT - 3.93e15T^{2} \)
97 \( 1 + 7.77e7T + 7.83e15T^{2} \)
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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

−14.35483339615164462062082996511, −12.79290166130959796764197958695, −12.16497700636294861068998409057, −10.98167143810102653725699971733, −8.890622495538315713105700091473, −7.85150738047728545441811895122, −6.47708459838371557937232429390, −5.59346384714061362630672613490, −2.68167760676288915695701240967, −1.46616431157382584197890701007, 1.09346899731175003901002504101, 3.30655716109600654747146163280, 4.32877846357322469051817660525, 6.43143149079570617133748469938, 7.76691600017287948219710921023, 9.803268462166043450822092114203, 10.61049417335860080235789005573, 11.21105283507710784597562120532, 12.99798754333866272721004383126, 14.22363502855677002935209912027

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