Properties

Label 2-140-140.19-c1-0-6
Degree $2$
Conductor $140$
Sign $0.919 - 0.392i$
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  + (−1.41 + 0.0377i)2-s + (0.634 − 0.366i)3-s + (1.99 − 0.106i)4-s + (−0.661 + 2.13i)5-s + (−0.883 + 0.542i)6-s + (2.56 − 0.664i)7-s + (−2.81 + 0.226i)8-s + (−1.23 + 2.13i)9-s + (0.853 − 3.04i)10-s + (2.33 − 1.34i)11-s + (1.22 − 0.799i)12-s + 3.95·13-s + (−3.59 + 1.03i)14-s + (0.363 + 1.59i)15-s + (3.97 − 0.426i)16-s + (0.709 + 1.22i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0267i)2-s + (0.366 − 0.211i)3-s + (0.998 − 0.0534i)4-s + (−0.295 + 0.955i)5-s + (−0.360 + 0.221i)6-s + (0.967 − 0.250i)7-s + (−0.996 + 0.0800i)8-s + (−0.410 + 0.710i)9-s + (0.270 − 0.962i)10-s + (0.702 − 0.405i)11-s + (0.354 − 0.230i)12-s + 1.09·13-s + (−0.960 + 0.276i)14-s + (0.0937 + 0.412i)15-s + (0.994 − 0.106i)16-s + (0.172 + 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.919 - 0.392i$
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: \((2,\ 140,\ (\ :1/2),\ 0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839243 + 0.171801i\)
\(L(\frac12)\) \(\approx\) \(0.839243 + 0.171801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0377i)T \)
5 \( 1 + (0.661 - 2.13i)T \)
7 \( 1 + (-2.56 + 0.664i)T \)
good3 \( 1 + (-0.634 + 0.366i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.95T + 13T^{2} \)
17 \( 1 + (-0.709 - 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.45 + 4.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 + (3.81 + 6.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.87 + 2.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.325iT - 41T^{2} \)
43 \( 1 + 9.28T + 43T^{2} \)
47 \( 1 + (5.68 + 3.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.39 + 0.807i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.81 - 6.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.3 - 7.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.51 - 2.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 + (0.709 + 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 + 6.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.26iT - 83T^{2} \)
89 \( 1 + (-4.10 - 2.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.35T + 97T^{2} \)
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

−13.36278584077456295052345669004, −11.66582752492963037035774975800, −11.08134520530873659930041798101, −10.33995873624195773810041425642, −8.735529457279304959843184390982, −8.099214102322601078995924949844, −7.10106060848917141365053437666, −5.88295141967872213834161489810, −3.60493064201504355915319827741, −1.93129246157356772913141571516, 1.47200268628992619524214957976, 3.63379558110419850192705742948, 5.35654287907033986554453660043, 6.88801301280767168272343000897, 8.285800008057652641398971997860, 8.818732851865302413872733685343, 9.622260365715832285460616343492, 11.23765610574099144031142018622, 11.70608211811142422023897652747, 12.86788439830382636445939561777

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