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Properties

Label 2-140-140.123-c1-0-18
Degree $2$
Conductor $140$
Sign $-0.568 + 0.822i$
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  + (0.764 − 1.18i)2-s + (−0.638 − 2.38i)3-s + (−0.831 − 1.81i)4-s + (2.14 + 0.631i)5-s + (−3.32 − 1.06i)6-s + (−1.60 + 2.10i)7-s + (−2.79 − 0.401i)8-s + (−2.68 + 1.54i)9-s + (2.39 − 2.06i)10-s + (4.09 + 2.36i)11-s + (−3.80 + 3.14i)12-s + (0.0592 + 0.0592i)13-s + (1.27 + 3.51i)14-s + (0.135 − 5.51i)15-s + (−2.61 + 3.02i)16-s + (−1.27 − 4.77i)17-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + (−0.368 − 1.37i)3-s + (−0.415 − 0.909i)4-s + (0.959 + 0.282i)5-s + (−1.35 − 0.433i)6-s + (−0.607 + 0.794i)7-s + (−0.989 − 0.142i)8-s + (−0.893 + 0.515i)9-s + (0.756 − 0.654i)10-s + (1.23 + 0.713i)11-s + (−1.09 + 0.907i)12-s + (0.0164 + 0.0164i)13-s + (0.339 + 0.940i)14-s + (0.0349 − 1.42i)15-s + (−0.654 + 0.755i)16-s + (−0.310 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.568 + 0.822i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.568 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618768 - 1.17923i\)
\(L(\frac12)\) \(\approx\) \(0.618768 - 1.17923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.764 + 1.18i)T \)
5 \( 1 + (-2.14 - 0.631i)T \)
7 \( 1 + (1.60 - 2.10i)T \)
good3 \( 1 + (0.638 + 2.38i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0592 - 0.0592i)T + 13iT^{2} \)
17 \( 1 + (1.27 + 4.77i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.31 + 2.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.09 - 0.292i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.27iT - 29T^{2} \)
31 \( 1 + (-4.01 - 2.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.22 - 0.596i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 + (1.57 - 1.57i)T - 43iT^{2} \)
47 \( 1 + (0.716 - 2.67i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.12 - 2.44i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.67 - 2.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.978 + 1.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.491 + 0.131i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (-9.26 + 2.48i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.05 - 5.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.62 - 5.62i)T - 83iT^{2} \)
89 \( 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.81 + 5.81i)T - 97iT^{2} \)
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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

−12.67203889004558567299752704033, −12.09956953448104838689433360525, −11.15325641462303577376573855597, −9.715613092008062582981202492874, −9.003003616014123706779843849048, −6.81501130291398716111201361909, −6.37101729498749811347038217714, −5.05420785223057603031048291774, −2.83002092834554401054380488250, −1.58560624449428496782729827499, 3.63626148797905064348225983888, 4.47065862429419359983795669489, 5.87259676074128143296757718214, 6.52195863290230953686839109958, 8.415454528878963440530535722495, 9.456809805447007361626984033278, 10.20848540013415643142094408408, 11.43083536191836382473349085276, 12.81349145103582744334852048032, 13.70643347183195719694081505321

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