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Properties

Label 2-140-7.6-c14-0-16
Degree $2$
Conductor $140$
Sign $0.0296 - 0.999i$
Analytic cond. $174.060$
Root an. cond. $13.1932$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 970. i·3-s − 3.49e4i·5-s + (8.23e5 + 2.44e4i)7-s + 3.84e6·9-s + 2.31e7·11-s − 2.07e7i·13-s + 3.38e7·15-s + 3.31e8i·17-s + 1.68e9i·19-s + (−2.37e7 + 7.98e8i)21-s − 5.02e9·23-s − 1.22e9·25-s + 8.36e9i·27-s + 3.08e10·29-s + 3.80e10i·31-s + ⋯
L(s)  = 1  + 0.443i·3-s − 0.447i·5-s + (0.999 + 0.0296i)7-s + 0.803·9-s + 1.18·11-s − 0.331i·13-s + 0.198·15-s + 0.809i·17-s + 1.88i·19-s + (−0.0131 + 0.443i)21-s − 1.47·23-s − 0.199·25-s + 0.799i·27-s + 1.78·29-s + 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0296 - 0.999i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.0296 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.0296 - 0.999i$
Analytic conductor: \(174.060\)
Root analytic conductor: \(13.1932\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :7),\ 0.0296 - 0.999i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(2.855402293\)
\(L(\frac12)\) \(\approx\) \(2.855402293\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 3.49e4iT \)
7 \( 1 + (-8.23e5 - 2.44e4i)T \)
good3 \( 1 - 970. iT - 4.78e6T^{2} \)
11 \( 1 - 2.31e7T + 3.79e14T^{2} \)
13 \( 1 + 2.07e7iT - 3.93e15T^{2} \)
17 \( 1 - 3.31e8iT - 1.68e17T^{2} \)
19 \( 1 - 1.68e9iT - 7.99e17T^{2} \)
23 \( 1 + 5.02e9T + 1.15e19T^{2} \)
29 \( 1 - 3.08e10T + 2.97e20T^{2} \)
31 \( 1 - 3.80e10iT - 7.56e20T^{2} \)
37 \( 1 + 1.07e11T + 9.01e21T^{2} \)
41 \( 1 + 2.01e11iT - 3.79e22T^{2} \)
43 \( 1 + 2.72e11T + 7.38e22T^{2} \)
47 \( 1 - 6.11e11iT - 2.56e23T^{2} \)
53 \( 1 - 5.86e11T + 1.37e24T^{2} \)
59 \( 1 + 4.45e12iT - 6.19e24T^{2} \)
61 \( 1 - 5.00e12iT - 9.87e24T^{2} \)
67 \( 1 + 4.06e12T + 3.67e25T^{2} \)
71 \( 1 - 1.17e13T + 8.27e25T^{2} \)
73 \( 1 - 5.21e12iT - 1.22e26T^{2} \)
79 \( 1 + 1.09e13T + 3.68e26T^{2} \)
83 \( 1 - 1.13e12iT - 7.36e26T^{2} \)
89 \( 1 - 3.12e13iT - 1.95e27T^{2} \)
97 \( 1 + 1.10e13iT - 6.52e27T^{2} \)
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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

−10.57044956466365560028587906919, −9.977752651546773401232303165386, −8.661264976752811847758696911275, −7.990333630719407543728386991778, −6.61485870102606637157535988004, −5.44037223647960072148028351019, −4.33143731165843914669586915860, −3.67818915370017756738803918419, −1.72696519622648641667021157816, −1.25209037502948227468392264438, 0.53706624199906073632947668958, 1.54667301494626410825923304731, 2.49488876524894254997362833582, 4.04221123645648704812806570109, 4.87266279653232465166788732770, 6.46851799872857962456813216278, 7.08547306171963656028322922575, 8.168612499751522221987997730021, 9.292186975850121291556745188789, 10.33805283689886661904828988203

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