Properties

Label 2-43-43.42-c4-0-5
Degree $2$
Conductor $43$
Sign $-0.0745 - 0.997i$
Analytic cond. $4.44490$
Root an. cond. $2.10829$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.133i·2-s + 14.8i·3-s + 15.9·4-s − 26.0i·5-s − 1.97·6-s + 87.9i·7-s + 4.26i·8-s − 138.·9-s + 3.47·10-s − 101.·11-s + 236. i·12-s + 190.·13-s − 11.7·14-s + 385.·15-s + 255.·16-s − 227.·17-s + ⋯
L(s)  = 1  + 0.0333i·2-s + 1.64i·3-s + 0.998·4-s − 1.04i·5-s − 0.0548·6-s + 1.79i·7-s + 0.0666i·8-s − 1.70·9-s + 0.0347·10-s − 0.840·11-s + 1.64i·12-s + 1.12·13-s − 0.0598·14-s + 1.71·15-s + 0.996·16-s − 0.788·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.0745 - 0.997i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ -0.0745 - 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.17485 + 1.26590i\)
\(L(\frac12)\) \(\approx\) \(1.17485 + 1.26590i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (137. + 1.84e3i)T \)
good2 \( 1 - 0.133iT - 16T^{2} \)
3 \( 1 - 14.8iT - 81T^{2} \)
5 \( 1 + 26.0iT - 625T^{2} \)
7 \( 1 - 87.9iT - 2.40e3T^{2} \)
11 \( 1 + 101.T + 1.46e4T^{2} \)
13 \( 1 - 190.T + 2.85e4T^{2} \)
17 \( 1 + 227.T + 8.35e4T^{2} \)
19 \( 1 + 304. iT - 1.30e5T^{2} \)
23 \( 1 - 797.T + 2.79e5T^{2} \)
29 \( 1 + 1.08e3iT - 7.07e5T^{2} \)
31 \( 1 - 433.T + 9.23e5T^{2} \)
37 \( 1 - 310. iT - 1.87e6T^{2} \)
41 \( 1 - 1.33e3T + 2.82e6T^{2} \)
47 \( 1 - 1.00e3T + 4.87e6T^{2} \)
53 \( 1 + 3.64e3T + 7.89e6T^{2} \)
59 \( 1 + 1.28e3T + 1.21e7T^{2} \)
61 \( 1 - 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.75e3T + 2.01e7T^{2} \)
71 \( 1 - 4.25e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.32e3iT - 2.83e7T^{2} \)
79 \( 1 + 227.T + 3.89e7T^{2} \)
83 \( 1 + 1.89e3T + 4.74e7T^{2} \)
89 \( 1 + 1.08e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.09e4T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.60118151551328094046641428777, −15.14143718400814314375438755455, −13.00695844548918930409503600616, −11.65890310106071817257822446312, −10.78457474043951332442525377739, −9.223846148068503352117825418777, −8.531536419778971310727590575771, −5.89833870847558574109162555497, −4.86267442932936355387105180397, −2.76516038077872148322488566760, 1.27544788562586107829633175294, 3.06927732898238882247780417775, 6.40067771180259219568478316355, 7.07465846830812508929073911249, 7.891037933038580423995626443439, 10.74471567353136241406644807939, 11.02484178992664422982967872940, 12.74312065542155527304198033460, 13.59329147859155880946878761708, 14.60420708558352966523831277305

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