Properties

Label 2-140-7.6-c14-0-5
Degree $2$
Conductor $140$
Sign $0.712 - 0.701i$
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  − 3.89e3i·3-s − 3.49e4i·5-s + (5.78e5 + 5.86e5i)7-s − 1.03e7·9-s + 3.17e7·11-s − 1.31e7i·13-s − 1.36e8·15-s + 6.48e8i·17-s − 1.38e9i·19-s + (2.28e9 − 2.25e9i)21-s − 2.14e9·23-s − 1.22e9·25-s + 2.18e10i·27-s − 1.35e10·29-s − 1.88e9i·31-s + ⋯
L(s)  = 1  − 1.78i·3-s − 0.447i·5-s + (0.701 + 0.712i)7-s − 2.17·9-s + 1.62·11-s − 0.209i·13-s − 0.796·15-s + 1.57i·17-s − 1.54i·19-s + (1.26 − 1.25i)21-s − 0.630·23-s − 0.199·25-s + 2.09i·27-s − 0.786·29-s − 0.0686i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.712 - 0.701i$
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: \((2,\ 140,\ (\ :7),\ 0.712 - 0.701i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.9573418047\)
\(L(\frac12)\) \(\approx\) \(0.9573418047\)
\(L(8)\) 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 \( 1 + (-5.78e5 - 5.86e5i)T \)
good3 \( 1 + 3.89e3iT - 4.78e6T^{2} \)
11 \( 1 - 3.17e7T + 3.79e14T^{2} \)
13 \( 1 + 1.31e7iT - 3.93e15T^{2} \)
17 \( 1 - 6.48e8iT - 1.68e17T^{2} \)
19 \( 1 + 1.38e9iT - 7.99e17T^{2} \)
23 \( 1 + 2.14e9T + 1.15e19T^{2} \)
29 \( 1 + 1.35e10T + 2.97e20T^{2} \)
31 \( 1 + 1.88e9iT - 7.56e20T^{2} \)
37 \( 1 - 9.85e10T + 9.01e21T^{2} \)
41 \( 1 - 2.92e11iT - 3.79e22T^{2} \)
43 \( 1 + 2.83e11T + 7.38e22T^{2} \)
47 \( 1 - 7.96e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.76e12T + 1.37e24T^{2} \)
59 \( 1 - 2.37e12iT - 6.19e24T^{2} \)
61 \( 1 - 1.38e12iT - 9.87e24T^{2} \)
67 \( 1 + 7.50e12T + 3.67e25T^{2} \)
71 \( 1 + 1.56e13T + 8.27e25T^{2} \)
73 \( 1 - 9.33e12iT - 1.22e26T^{2} \)
79 \( 1 + 1.71e13T + 3.68e26T^{2} \)
83 \( 1 + 2.70e12iT - 7.36e26T^{2} \)
89 \( 1 - 7.66e13iT - 1.95e27T^{2} \)
97 \( 1 - 1.58e13iT - 6.52e27T^{2} \)
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

−11.23438658496863485189690002207, −9.250876629047041633942446701586, −8.469046561772567876755534201142, −7.67223017315927089568807457024, −6.46672311560732971062989000061, −5.88201485688313463155093954526, −4.39446165634382010517635714970, −2.76130704162639216500690622997, −1.56260606810583465553103541390, −1.25314632720691526818037963367, 0.17096919903993937079039443193, 1.73055599524049541736824233679, 3.38711830229734076649109843854, 3.99576610974136934603167248769, 4.86599844224863607466777616622, 6.05345965983773292732852521504, 7.40274418018139884973912727011, 8.720891286156042730691509280946, 9.639038500498643130326815822142, 10.28540497629857360169323563905

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