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Properties

Label 2-140-140.27-c2-0-13
Degree $2$
Conductor $140$
Sign $0.976 - 0.214i$
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  + (−0.231 − 1.98i)2-s + (−0.780 + 0.780i)3-s + (−3.89 + 0.919i)4-s + (4.66 + 1.78i)5-s + (1.73 + 1.36i)6-s + (−1.41 + 6.85i)7-s + (2.72 + 7.52i)8-s + 7.78i·9-s + (2.47 − 9.68i)10-s − 6.47i·11-s + (2.32 − 3.75i)12-s + (5.86 + 5.86i)13-s + (13.9 + 1.21i)14-s + (−5.03 + 2.24i)15-s + (14.3 − 7.15i)16-s + (19.7 − 19.7i)17-s + ⋯
L(s)  = 1  + (−0.115 − 0.993i)2-s + (−0.260 + 0.260i)3-s + (−0.973 + 0.229i)4-s + (0.933 + 0.357i)5-s + (0.288 + 0.228i)6-s + (−0.201 + 0.979i)7-s + (0.340 + 0.940i)8-s + 0.864i·9-s + (0.247 − 0.968i)10-s − 0.588i·11-s + (0.193 − 0.312i)12-s + (0.451 + 0.451i)13-s + (0.996 + 0.0871i)14-s + (−0.335 + 0.149i)15-s + (0.894 − 0.447i)16-s + (1.16 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.976 - 0.214i$
Analytic conductor: \(3.81472\)
Root analytic conductor: \(1.95313\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1),\ 0.976 - 0.214i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.23147 + 0.133688i\)
\(L(\frac12)\) \(\approx\) \(1.23147 + 0.133688i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.231 + 1.98i)T \)
5 \( 1 + (-4.66 - 1.78i)T \)
7 \( 1 + (1.41 - 6.85i)T \)
good3 \( 1 + (0.780 - 0.780i)T - 9iT^{2} \)
11 \( 1 + 6.47iT - 121T^{2} \)
13 \( 1 + (-5.86 - 5.86i)T + 169iT^{2} \)
17 \( 1 + (-19.7 + 19.7i)T - 289iT^{2} \)
19 \( 1 - 33.6iT - 361T^{2} \)
23 \( 1 + (14.4 + 14.4i)T + 529iT^{2} \)
29 \( 1 - 26.3iT - 841T^{2} \)
31 \( 1 - 4.12T + 961T^{2} \)
37 \( 1 + (40.5 + 40.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 45.5iT - 1.68e3T^{2} \)
43 \( 1 + (-0.647 - 0.647i)T + 1.84e3iT^{2} \)
47 \( 1 + (-40.2 - 40.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-9.95 + 9.95i)T - 2.80e3iT^{2} \)
59 \( 1 + 66.9iT - 3.48e3T^{2} \)
61 \( 1 + 32.6iT - 3.72e3T^{2} \)
67 \( 1 + (-69.6 + 69.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 1.25iT - 5.04e3T^{2} \)
73 \( 1 + (53.6 + 53.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 39.7T + 6.24e3T^{2} \)
83 \( 1 + (-102. + 102. i)T - 6.88e3iT^{2} \)
89 \( 1 - 12.7T + 7.92e3T^{2} \)
97 \( 1 + (-21.3 + 21.3i)T - 9.40e3iT^{2} \)
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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

−12.74422499970380255612960161911, −11.88158593627016403530168325610, −10.82321989958487779685508033488, −10.02804865123220492823073067654, −9.141932525509414758040211999561, −7.983107485470744285750345912567, −5.97460898590301945960731315702, −5.14185304816936011328684721442, −3.27663057255615972110613637234, −1.91528706406716151920641908976, 0.983564515303056654631523879171, 3.90366956735165512507633753421, 5.39800518650533897359268243021, 6.38983193012437081513962873618, 7.29956876445826234907668871887, 8.626891216875934442356610380606, 9.723080631301649500505198150922, 10.42194551162611059330665145325, 12.21656361357741041577793917665, 13.21250196038152739260747272567

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