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Properties

Label 2-42-21.20-c27-0-54
Degree $2$
Conductor $42$
Sign $0.634 + 0.772i$
Analytic cond. $193.979$
Root an. cond. $13.9276$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.19e3i·2-s + (−2.76e6 + 7.28e4i)3-s − 6.71e7·4-s + 5.43e9·5-s + (5.96e8 + 2.26e10i)6-s + (2.02e11 − 1.57e11i)7-s + 5.49e11i·8-s + (7.61e12 − 4.02e11i)9-s − 4.45e13i·10-s + 1.42e14i·11-s + (1.85e14 − 4.88e12i)12-s + 1.72e15i·13-s + (−1.28e15 − 1.65e15i)14-s + (−1.50e16 + 3.96e14i)15-s + 4.50e15·16-s + 3.48e16·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.999 + 0.0263i)3-s − 0.5·4-s + 1.99·5-s + (0.0186 + 0.706i)6-s + (0.789 − 0.614i)7-s + 0.353i·8-s + (0.998 − 0.0527i)9-s − 1.40i·10-s + 1.24i·11-s + (0.499 − 0.0131i)12-s + 1.57i·13-s + (−0.434 − 0.558i)14-s + (−1.99 + 0.0525i)15-s + 0.250·16-s + 0.854·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(193.979\)
Root analytic conductor: \(13.9276\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :27/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(14)\) \(\approx\) \(3.053549971\)
\(L(\frac12)\) \(\approx\) \(3.053549971\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8.19e3iT \)
3 \( 1 + (2.76e6 - 7.28e4i)T \)
7 \( 1 + (-2.02e11 + 1.57e11i)T \)
good5 \( 1 - 5.43e9T + 7.45e18T^{2} \)
11 \( 1 - 1.42e14iT - 1.31e28T^{2} \)
13 \( 1 - 1.72e15iT - 1.19e30T^{2} \)
17 \( 1 - 3.48e16T + 1.66e33T^{2} \)
19 \( 1 + 9.58e16iT - 3.36e34T^{2} \)
23 \( 1 + 3.80e18iT - 5.84e36T^{2} \)
29 \( 1 + 3.14e19iT - 3.05e39T^{2} \)
31 \( 1 + 4.59e19iT - 1.84e40T^{2} \)
37 \( 1 + 1.14e20T + 2.19e42T^{2} \)
41 \( 1 + 8.22e20T + 3.50e43T^{2} \)
43 \( 1 + 6.76e20T + 1.26e44T^{2} \)
47 \( 1 - 4.92e22T + 1.40e45T^{2} \)
53 \( 1 + 1.33e23iT - 3.59e46T^{2} \)
59 \( 1 + 7.27e23T + 6.50e47T^{2} \)
61 \( 1 - 9.98e22iT - 1.59e48T^{2} \)
67 \( 1 + 3.30e24T + 2.01e49T^{2} \)
71 \( 1 - 9.60e24iT - 9.63e49T^{2} \)
73 \( 1 + 1.60e25iT - 2.04e50T^{2} \)
79 \( 1 + 1.96e25T + 1.72e51T^{2} \)
83 \( 1 - 4.28e24T + 6.53e51T^{2} \)
89 \( 1 - 2.31e26T + 4.30e52T^{2} \)
97 \( 1 + 8.59e26iT - 4.39e53T^{2} \)
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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

−10.60781445659192988397357405229, −10.00502322631303163094136435168, −9.118055389608993895182902868862, −7.12597810832621567006372383824, −6.17276692747449340891465486625, −4.95512300231353781952142169655, −4.38446600916312794183956362628, −2.23826410438476155121763366786, −1.70478564183781986336329602035, −0.807455682717634601686091509090, 0.879862574599709975290454844177, 1.57698358813928369207176264901, 3.11213081722714753674424489270, 5.16417890044058393265055260708, 5.63237791226233907487616915279, 6.05962005318798320338241223792, 7.62504375602215785432314865706, 8.917162896138910950056150339918, 10.04748882944646099608616388428, 10.85301150204266993407128351474

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