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Properties

Label 2-140-28.19-c1-0-2
Degree $2$
Conductor $140$
Sign $0.941 + 0.337i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 0.242i)2-s + (−0.406 − 0.703i)3-s + (1.88 + 0.675i)4-s + (0.866 + 0.5i)5-s + (0.395 + 1.07i)6-s + (0.336 + 2.62i)7-s + (−2.45 − 1.39i)8-s + (1.17 − 2.02i)9-s + (−1.08 − 0.906i)10-s + (4.20 − 2.43i)11-s + (−0.289 − 1.59i)12-s + 0.895i·13-s + (0.167 − 3.73i)14-s − 0.812i·15-s + (3.08 + 2.54i)16-s + (5.10 − 2.94i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.171i)2-s + (−0.234 − 0.406i)3-s + (0.941 + 0.337i)4-s + (0.387 + 0.223i)5-s + (0.161 + 0.440i)6-s + (0.127 + 0.991i)7-s + (−0.869 − 0.494i)8-s + (0.390 − 0.675i)9-s + (−0.343 − 0.286i)10-s + (1.26 − 0.732i)11-s + (−0.0835 − 0.461i)12-s + 0.248i·13-s + (0.0447 − 0.998i)14-s − 0.209i·15-s + (0.771 + 0.635i)16-s + (1.23 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.941 + 0.337i$
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: \((2,\ 140,\ (\ :1/2),\ 0.941 + 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773122 - 0.134208i\)
\(L(\frac12)\) \(\approx\) \(0.773122 - 0.134208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.242i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.336 - 2.62i)T \)
good3 \( 1 + (0.406 + 0.703i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-4.20 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.895iT - 13T^{2} \)
17 \( 1 + (-5.10 + 2.94i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.45 - 2.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.35 + 0.780i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.73T + 29T^{2} \)
31 \( 1 + (-2.20 - 3.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.910 + 1.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.4iT - 41T^{2} \)
43 \( 1 + 3.04iT - 43T^{2} \)
47 \( 1 + (2.60 - 4.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0898 + 0.155i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.68 + 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.33 + 3.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.75 + 3.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.38iT - 71T^{2} \)
73 \( 1 + (7.60 - 4.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.14 - 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 + (-1.52 - 0.882i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.83iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.65721687520920807340389140925, −11.93019688109467531826951169777, −11.20762001970171975438245619660, −9.698592178541626413082587850218, −9.169458805020578220326068236285, −7.918648029521631263513357061756, −6.61213502881845437279653718558, −5.86862035651934035847648105955, −3.36841154945088229348773730915, −1.54794030095181135886005846946, 1.61337056537262084678569422022, 4.06529257105259183784828318708, 5.63878008421618350750169396754, 7.00593373415617851305701708128, 7.88390633338020426113865139764, 9.300767707989347745653504129371, 10.07142328059136173771651076673, 10.82027824256447039708854007081, 11.92665381673023653352630812738, 13.19699837280164665906163049334

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