↑

Properties

Label 2-140-140.19-c1-0-18
Degree $2$
Conductor $140$
Sign $-0.255 + 0.966i$
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  + (0.163 − 1.40i)2-s + (1.62 − 0.940i)3-s + (−1.94 − 0.458i)4-s + (0.430 − 2.19i)5-s + (−1.05 − 2.44i)6-s + (−0.603 + 2.57i)7-s + (−0.960 + 2.66i)8-s + (0.267 − 0.463i)9-s + (−3.01 − 0.962i)10-s + (−1.20 + 0.692i)11-s + (−3.60 + 1.08i)12-s + 5.79·13-s + (3.52 + 1.26i)14-s + (−1.36 − 3.97i)15-s + (3.58 + 1.78i)16-s + (−2.33 − 4.04i)17-s + ⋯
L(s)  = 1  + (0.115 − 0.993i)2-s + (0.940 − 0.542i)3-s + (−0.973 − 0.229i)4-s + (0.192 − 0.981i)5-s + (−0.430 − 0.996i)6-s + (−0.228 + 0.973i)7-s + (−0.339 + 0.940i)8-s + (0.0891 − 0.154i)9-s + (−0.952 − 0.304i)10-s + (−0.361 + 0.208i)11-s + (−1.03 + 0.313i)12-s + 1.60·13-s + (0.940 + 0.338i)14-s + (−0.351 − 1.02i)15-s + (0.895 + 0.445i)16-s + (−0.565 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.255 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835791 - 1.08530i\)
\(L(\frac12)\) \(\approx\) \(0.835791 - 1.08530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.163 + 1.40i)T \)
5 \( 1 + (-0.430 + 2.19i)T \)
7 \( 1 + (0.603 - 2.57i)T \)
good3 \( 1 + (-1.62 + 0.940i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.20 - 0.692i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 + (2.33 + 4.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.74 - 4.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.04T + 29T^{2} \)
31 \( 1 + (-0.832 - 1.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.247 - 0.142i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.21iT - 41T^{2} \)
43 \( 1 + 3.30T + 43T^{2} \)
47 \( 1 + (0.119 + 0.0690i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.82 - 3.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.832 + 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.49 + 1.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.82 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 + (-2.33 - 4.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.88 + 5.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.0814iT - 83T^{2} \)
89 \( 1 + (3.22 + 1.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.88T + 97T^{2} \)
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

−12.94309018023471037387836169820, −12.12607905284186611880302514446, −10.94800831611404443094600021933, −9.559886595606225337072494958232, −8.650137385516643002878527581651, −8.246742151312829091992580967367, −6.04786755375808907904511413012, −4.71241228055399277622967285238, −3.04285779389843483432689960030, −1.74261498094097064639484404636, 3.28840205197568346941623049810, 4.13757507393546601714498190502, 6.06903815702748982521113162689, 6.97402453205548680432720658692, 8.213256518634609197613810872487, 9.062159721863065800599127742124, 10.22382214549423037548998061265, 11.07495320939933886392832728584, 13.22568462771932809857623675063, 13.59868382717553084979267573969

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