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Properties

Label 2-140-140.19-c1-0-10
Degree $2$
Conductor $140$
Sign $0.623 - 0.781i$
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  + (0.674 + 1.24i)2-s + (0.634 − 0.366i)3-s + (−1.09 + 1.67i)4-s + (1.51 − 1.64i)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 + (3.06 + 0.782i)10-s + (−2.33 + 1.34i)11-s + (−0.0783 + 1.46i)12-s − 3.95·13-s + (2.55 + 2.73i)14-s + (0.363 − 1.59i)15-s + (−1.61 − 3.65i)16-s + (−0.709 − 1.22i)17-s + ⋯
L(s)  = 1  + (0.476 + 0.879i)2-s + (0.366 − 0.211i)3-s + (−0.545 + 0.838i)4-s + (0.679 − 0.733i)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 + (0.968 + 0.247i)10-s + (−0.702 + 0.405i)11-s + (−0.0226 + 0.422i)12-s − 1.09·13-s + (0.682 + 0.731i)14-s + (0.0937 − 0.412i)15-s + (−0.404 − 0.914i)16-s + (−0.172 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.623 - 0.781i$
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: \((2,\ 140,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38374 + 0.666077i\)
\(L(\frac12)\) \(\approx\) \(1.38374 + 0.666077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.674 - 1.24i)T \)
5 \( 1 + (-1.51 + 1.64i)T \)
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 \( 1 - 8.35T + 97T^{2} \)
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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

−13.43140208278965574903122955551, −12.74306339934060148255413768447, −11.52433828789139527107562860109, −10.01306706608514889058587993862, −8.747018100543254262155291671936, −7.976866267718721047919129363358, −6.96259355015795476040414610002, −5.15480085763048935182899262077, −4.85848101523503827104728922014, −2.48146992367549406482406576691, 2.17844869790961090157119686010, 3.39004094528068690656285111497, 5.06736425530333917755169001787, 6.07661176286011546048777875475, 7.85255189518772186474989492525, 9.249923506898875076781585179437, 10.00213375971598218524540118583, 11.11359899623420890150440043798, 11.82666818892462566860675577687, 13.10430540206303043391108199609

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