↑

Properties

Label 2-140-7.6-c14-0-6
Degree $2$
Conductor $140$
Sign $0.577 - 0.816i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e3i·3-s + 3.49e4i·5-s + (−6.72e5 − 4.75e5i)7-s + 5.99e5·9-s + 2.05e7·11-s − 6.03e7i·13-s + 7.14e7·15-s + 3.82e8i·17-s + 8.94e8i·19-s + (−9.72e8 + 1.37e9i)21-s − 1.78e9·23-s − 1.22e9·25-s − 1.10e10i·27-s + 8.73e9·29-s + 1.20e10i·31-s + ⋯
L(s)  = 1  − 0.935i·3-s + 0.447i·5-s + (−0.816 − 0.577i)7-s + 0.125·9-s + 1.05·11-s − 0.961i·13-s + 0.418·15-s + 0.932i·17-s + 1.00i·19-s + (−0.540 + 0.763i)21-s − 0.525·23-s − 0.199·25-s − 1.05i·27-s + 0.506·29-s + 0.438i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.577 - 0.816i$
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: \((2,\ 140,\ (\ :7),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.224689281\)
\(L(\frac12)\) \(\approx\) \(1.224689281\)
\(L(8)\) 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 \( 1 + (6.72e5 + 4.75e5i)T \)
good3 \( 1 + 2.04e3iT - 4.78e6T^{2} \)
11 \( 1 - 2.05e7T + 3.79e14T^{2} \)
13 \( 1 + 6.03e7iT - 3.93e15T^{2} \)
17 \( 1 - 3.82e8iT - 1.68e17T^{2} \)
19 \( 1 - 8.94e8iT - 7.99e17T^{2} \)
23 \( 1 + 1.78e9T + 1.15e19T^{2} \)
29 \( 1 - 8.73e9T + 2.97e20T^{2} \)
31 \( 1 - 1.20e10iT - 7.56e20T^{2} \)
37 \( 1 + 1.08e11T + 9.01e21T^{2} \)
41 \( 1 - 2.46e10iT - 3.79e22T^{2} \)
43 \( 1 + 1.01e10T + 7.38e22T^{2} \)
47 \( 1 + 4.26e11iT - 2.56e23T^{2} \)
53 \( 1 + 4.19e11T + 1.37e24T^{2} \)
59 \( 1 - 7.09e11iT - 6.19e24T^{2} \)
61 \( 1 + 3.13e11iT - 9.87e24T^{2} \)
67 \( 1 + 8.39e12T + 3.67e25T^{2} \)
71 \( 1 + 1.41e13T + 8.27e25T^{2} \)
73 \( 1 - 1.11e13iT - 1.22e26T^{2} \)
79 \( 1 - 3.24e12T + 3.68e26T^{2} \)
83 \( 1 - 1.83e13iT - 7.36e26T^{2} \)
89 \( 1 - 4.09e13iT - 1.95e27T^{2} \)
97 \( 1 + 6.16e12iT - 6.52e27T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.53421384353759537658429378893, −9.935521492513386445084767482659, −8.480054739370320724008736500212, −7.45908049264831020319890575479, −6.63038902323681776669866653456, −5.90678388751424980018781971241, −4.08085868002279452366834435679, −3.22356683263277810864532220451, −1.79871175413979427733229163624, −0.945176886598133112356208072683, 0.25703010036622631917942095588, 1.65269847394483989799119153911, 3.03093658420029207354104471609, 4.12722091896090599255409754029, 4.89695477208128390531340695296, 6.20777322932931327216706956482, 7.16677565652559837609861381251, 8.970227990419364413077590620576, 9.231884250222249082174647906822, 10.17054352458045294391604494532

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