Properties

Degree 2
Conductor 43
Sign $-0.931 + 0.364i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.9i·2-s − 41.5i·3-s − 77.9·4-s + 111. i·5-s + 495.·6-s + 49.3i·7-s − 166. i·8-s − 998.·9-s − 1.32e3·10-s − 2.54e3·11-s + 3.24e3i·12-s − 277.·13-s − 587.·14-s + 4.63e3·15-s − 3.00e3·16-s − 532.·17-s + ⋯
L(s)  = 1  + 1.48i·2-s − 1.53i·3-s − 1.21·4-s + 0.892i·5-s + 2.29·6-s + 0.143i·7-s − 0.325i·8-s − 1.36·9-s − 1.32·10-s − 1.91·11-s + 1.87i·12-s − 0.126·13-s − 0.214·14-s + 1.37·15-s − 0.733·16-s − 0.108·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.931 + 0.364i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.931 + 0.364i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0956478 - 0.506582i\)
\(L(\frac12)\)  \(\approx\)  \(0.0956478 - 0.506582i\)
\(L(4)\)   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 \( 1 + (-7.40e4 + 2.89e4i)T \)
good2 \( 1 - 11.9iT - 64T^{2} \)
3 \( 1 + 41.5iT - 729T^{2} \)
5 \( 1 - 111. iT - 1.56e4T^{2} \)
7 \( 1 - 49.3iT - 1.17e5T^{2} \)
11 \( 1 + 2.54e3T + 1.77e6T^{2} \)
13 \( 1 + 277.T + 4.82e6T^{2} \)
17 \( 1 + 532.T + 2.41e7T^{2} \)
19 \( 1 - 6.00e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.85e4T + 1.48e8T^{2} \)
29 \( 1 - 2.70e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.61e4T + 8.87e8T^{2} \)
37 \( 1 + 2.17e3iT - 2.56e9T^{2} \)
41 \( 1 + 8.26e4T + 4.75e9T^{2} \)
47 \( 1 + 1.68e4T + 1.07e10T^{2} \)
53 \( 1 + 8.04e4T + 2.21e10T^{2} \)
59 \( 1 - 1.51e5T + 4.21e10T^{2} \)
61 \( 1 - 3.48e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.61e5T + 9.04e10T^{2} \)
71 \( 1 + 4.95e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.04e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.89e5T + 2.43e11T^{2} \)
83 \( 1 - 4.35e5T + 3.26e11T^{2} \)
89 \( 1 - 1.28e6iT - 4.96e11T^{2} \)
97 \( 1 + 2.91e5T + 8.32e11T^{2} \)
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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

−15.30718969012111741532678425963, −14.19618849400912782059295111069, −13.40945074303952024939658716419, −12.18422749072527676330205749609, −10.50267200859393755538902265304, −8.251895574245341395712496365918, −7.57632013563548740075652038288, −6.59693539430850476168865389937, −5.51497287460087394544791300464, −2.42183820657512897667728424528, 0.22607126320321661333934267650, 2.62946233664440245287316400799, 4.20093461180804021845925147401, 5.14813611496041841008526023422, 8.370826086087888199699643164888, 9.692914755822400801118564028727, 10.32654796837854653214683044334, 11.34495102406490405356618339013, 12.65104043474008496114012400393, 13.67057317417679786316006921088

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