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Properties

Label 2-140-35.27-c1-0-2
Degree $2$
Conductor $140$
Sign $0.648 + 0.761i$
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  + (−0.386 − 0.386i)3-s + (−0.386 − 2.20i)5-s + (2.64 + 0.0564i)7-s − 2.70i·9-s + 1.70·11-s + (−0.386 − 0.386i)13-s + (−0.701 + i)15-s + (−4.79 + 4.79i)17-s + 5.95·19-s + (−0.999 − 1.04i)21-s + (−2.70 + 2.70i)23-s + (−4.70 + 1.70i)25-s + (−2.20 + 2.20i)27-s − 5.70i·29-s + 8.03i·31-s + ⋯
L(s)  = 1  + (−0.223 − 0.223i)3-s + (−0.172 − 0.984i)5-s + (0.999 + 0.0213i)7-s − 0.900i·9-s + 0.513·11-s + (−0.107 − 0.107i)13-s + (−0.181 + 0.258i)15-s + (−1.16 + 1.16i)17-s + 1.36·19-s + (−0.218 − 0.227i)21-s + (−0.563 + 0.563i)23-s + (−0.940 + 0.340i)25-s + (−0.423 + 0.423i)27-s − 1.05i·29-s + 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.648 + 0.761i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.648 + 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978329 - 0.451599i\)
\(L(\frac12)\) \(\approx\) \(0.978329 - 0.451599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.386 + 2.20i)T \)
7 \( 1 + (-2.64 - 0.0564i)T \)
good3 \( 1 + (0.386 + 0.386i)T + 3iT^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 + (0.386 + 0.386i)T + 13iT^{2} \)
17 \( 1 + (4.79 - 4.79i)T - 17iT^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 + (2.70 - 2.70i)T - 23iT^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 + (-2.70 - 2.70i)T + 37iT^{2} \)
41 \( 1 - 5.95iT - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (-3.24 + 3.24i)T - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 5.95T + 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + (-1.81 - 1.81i)T + 73iT^{2} \)
79 \( 1 + 0.298iT - 79T^{2} \)
83 \( 1 + (4.13 + 4.13i)T + 83iT^{2} \)
89 \( 1 + 2.08T + 89T^{2} \)
97 \( 1 + (-1.15 + 1.15i)T - 97iT^{2} \)
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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

−12.89803833010202207892595329296, −11.91377738992548057992753253779, −11.37046598625176999205064771439, −9.788290033268483861510536193498, −8.764082800839908858826243388474, −7.86326328554628660736260398184, −6.41905764540808110007188986660, −5.14458802819963715228890735626, −3.92713388405506045902296476051, −1.41588156006724541351922858563, 2.38766482791972815346043594420, 4.23585798116785910336953872767, 5.45183447578942555857929307049, 6.99750708504876954512138155807, 7.82306155095197290979609090998, 9.220145550570990635544843835953, 10.48799287663772018471069862953, 11.25983276443007704165914067511, 11.89791258155576416475834691849, 13.67902949669992686087129919173

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