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Properties

Label 2-140-7.4-c1-0-1
Degree $2$
Conductor $140$
Sign $0.605 + 0.795i$
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  + (1.5 − 2.59i)3-s + (0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−3 − 5.19i)9-s + (1 − 1.73i)11-s − 6·13-s + 3·15-s + (−1 + 1.73i)17-s + (7.5 + 2.59i)21-s + (4.5 + 7.79i)23-s + (−0.499 + 0.866i)25-s − 9·27-s + 3·29-s + (−1 + 1.73i)31-s + (−3 − 5.19i)33-s + ⋯
L(s)  = 1  + (0.866 − 1.49i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−1 − 1.73i)9-s + (0.301 − 0.522i)11-s − 1.66·13-s + 0.774·15-s + (−0.242 + 0.420i)17-s + (1.63 + 0.566i)21-s + (0.938 + 1.62i)23-s + (−0.0999 + 0.173i)25-s − 1.73·27-s + 0.557·29-s + (−0.179 + 0.311i)31-s + (−0.522 − 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24404 - 0.616705i\)
\(L(\frac12)\) \(\approx\) \(1.24404 - 0.616705i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good3 \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + T + 83T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.94793081897094035412996021100, −12.26705420246019802564831307460, −11.32006546482095040395029376052, −9.567992273346978297347743560120, −8.696057970109697825872006184581, −7.61817640236334635123069039669, −6.76744508605909680024719185851, −5.49297177427976641054165719345, −3.06969582227842589892302745920, −1.95938077675247174214824466243, 2.73897840237183253770951264479, 4.38522254492250913918981177806, 4.86887653197319161164792302111, 7.04975275008661300422987347886, 8.299185265549282275363671131429, 9.406135851580823636168006377522, 10.02227595596695412511737405192, 10.89001663872321024420385947457, 12.34319715475179651851535376233, 13.60122216544441798159624896179

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