↑

Properties

Label 2-140-28.19-c1-0-7
Degree $2$
Conductor $140$
Sign $0.822 + 0.569i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.501 + 1.32i)2-s + (−0.895 − 1.55i)3-s + (−1.49 − 1.32i)4-s + (0.866 + 0.5i)5-s + (2.49 − 0.405i)6-s + (−0.644 − 2.56i)7-s + (2.50 − 1.31i)8-s + (−0.103 + 0.179i)9-s + (−1.09 + 0.894i)10-s + (3.66 − 2.11i)11-s + (−0.717 + 3.50i)12-s − 2.98i·13-s + (3.71 + 0.434i)14-s − 1.79i·15-s + (0.480 + 3.97i)16-s + (−1.92 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.354 + 0.934i)2-s + (−0.516 − 0.895i)3-s + (−0.748 − 0.663i)4-s + (0.387 + 0.223i)5-s + (1.02 − 0.165i)6-s + (−0.243 − 0.969i)7-s + (0.885 − 0.464i)8-s + (−0.0344 + 0.0596i)9-s + (−0.346 + 0.282i)10-s + (1.10 − 0.638i)11-s + (−0.207 + 1.01i)12-s − 0.827i·13-s + (0.993 + 0.116i)14-s − 0.462i·15-s + (0.120 + 0.992i)16-s + (−0.465 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735656 - 0.229902i\)
\(L(\frac12)\) \(\approx\) \(0.735656 - 0.229902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.501 - 1.32i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.644 + 2.56i)T \)
good3 \( 1 + (0.895 + 1.55i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-3.66 + 2.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.98iT - 13T^{2} \)
17 \( 1 + (1.92 - 1.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.28 - 3.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.78 + 1.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + (-1.20 - 2.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.16 + 3.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.88iT - 41T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + (-3.38 + 5.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.99 - 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0195 - 0.0113i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.38 + 2.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.07iT - 71T^{2} \)
73 \( 1 + (2.88 - 1.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.14 + 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (14.4 + 8.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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

−13.27486830845068431403431458109, −12.27199381812773375954270536427, −10.81872373511511139228421692468, −9.958811122531885344285258774378, −8.636965013076370411934651814441, −7.47224273632261544926297982805, −6.51338532341866743644675805845, −5.93235815690483465018803923134, −4.05452377299971480757075151577, −1.05125917738979327417903620168, 2.19581218299141792295966234456, 4.12654000811707338000284151119, 5.05116053571708070744855867607, 6.66267458011703046201351975586, 8.603960552101396376822303197169, 9.414462964267419734421238061711, 10.04931076746415016214874678667, 11.33340342927075345492910495889, 11.89950495971581256063659278781, 12.97790636574160106440532855589

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