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Properties

Label 2-140-28.23-c2-0-18
Degree $2$
Conductor $140$
Sign $0.505 + 0.862i$
Analytic cond. $3.81472$
Root an. cond. $1.95313$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 − 1.07i)2-s + (−4.93 + 2.84i)3-s + (1.70 − 3.61i)4-s + (1.11 − 1.93i)5-s + (−5.28 + 10.0i)6-s + (6.48 − 2.62i)7-s + (−0.981 − 7.93i)8-s + (11.7 − 20.3i)9-s + (−0.183 − 4.46i)10-s + (5.26 − 3.03i)11-s + (1.86 + 22.7i)12-s + 11.5·13-s + (8.14 − 11.3i)14-s + 12.7i·15-s + (−10.1 − 12.3i)16-s + (2.84 + 4.92i)17-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−1.64 + 0.949i)3-s + (0.427 − 0.904i)4-s + (0.223 − 0.387i)5-s + (−0.881 + 1.68i)6-s + (0.926 − 0.375i)7-s + (−0.122 − 0.992i)8-s + (1.30 − 2.25i)9-s + (−0.0183 − 0.446i)10-s + (0.478 − 0.276i)11-s + (0.155 + 1.89i)12-s + 0.888·13-s + (0.582 − 0.813i)14-s + 0.849i·15-s + (−0.634 − 0.772i)16-s + (0.167 + 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(3.81472\)
Root analytic conductor: \(1.95313\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34840 - 0.772579i\)
\(L(\frac12)\) \(\approx\) \(1.34840 - 0.772579i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.68 + 1.07i)T \)
5 \( 1 + (-1.11 + 1.93i)T \)
7 \( 1 + (-6.48 + 2.62i)T \)
good3 \( 1 + (4.93 - 2.84i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (-5.26 + 3.03i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 11.5T + 169T^{2} \)
17 \( 1 + (-2.84 - 4.92i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (29.1 + 16.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.6 - 10.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 10.7T + 841T^{2} \)
31 \( 1 + (-14.2 + 8.24i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (20.3 - 35.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 41.1T + 1.68e3T^{2} \)
43 \( 1 - 6.10iT - 1.84e3T^{2} \)
47 \( 1 + (12.6 + 7.30i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.0 - 26.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.1 + 16.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.4 - 70.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.33 + 4.23i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 21.6iT - 5.04e3T^{2} \)
73 \( 1 + (13.0 + 22.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (7.65 + 4.42i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 + (26.6 - 46.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 176.T + 9.40e3T^{2} \)
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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

−12.53827204616940139877750321148, −11.45498965095317174098835303672, −11.04150568949782871641562643560, −10.22948758093856490572199107218, −8.963596777738745221391472427512, −6.68292346730201840119105879228, −5.75909958915055996609053381767, −4.75169967676470937762669462134, −3.97726007201447975998914316073, −1.10425811383229863660516954061, 1.87063135611341948904496931832, 4.42804546073876625007630655494, 5.60144479271463571970859758921, 6.33661706235802859456203765090, 7.25923859625912214591716137528, 8.409038922177299618610178187781, 10.72655311904683966607808932104, 11.27887746755649209925199429694, 12.24087601008992387396216177467, 12.86371994679210759884227739102

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