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Properties

Label 2-140-140.27-c2-0-11
Degree $2$
Conductor $140$
Sign $-0.996 + 0.0860i$
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  + (1.25 + 1.55i)2-s + (−3.43 + 3.43i)3-s + (−0.866 + 3.90i)4-s + (2.07 + 4.54i)5-s + (−9.64 − 1.05i)6-s + (4.38 − 5.45i)7-s + (−7.17 + 3.53i)8-s − 14.5i·9-s + (−4.49 + 8.93i)10-s + 9.86i·11-s + (−10.4 − 16.3i)12-s + (−8.85 − 8.85i)13-s + (13.9 + 0.0102i)14-s + (−22.7 − 8.48i)15-s + (−14.4 − 6.77i)16-s + (15.3 − 15.3i)17-s + ⋯
L(s)  = 1  + (0.625 + 0.779i)2-s + (−1.14 + 1.14i)3-s + (−0.216 + 0.976i)4-s + (0.414 + 0.909i)5-s + (−1.60 − 0.176i)6-s + (0.626 − 0.779i)7-s + (−0.897 + 0.441i)8-s − 1.61i·9-s + (−0.449 + 0.893i)10-s + 0.897i·11-s + (−0.868 − 1.36i)12-s + (−0.681 − 0.681i)13-s + (0.999 + 0.000732i)14-s + (−1.51 − 0.565i)15-s + (−0.906 − 0.423i)16-s + (0.902 − 0.902i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.996 + 0.0860i$
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: \((2,\ 140,\ (\ :1),\ -0.996 + 0.0860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0567408 - 1.31645i\)
\(L(\frac12)\) \(\approx\) \(0.0567408 - 1.31645i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 1.55i)T \)
5 \( 1 + (-2.07 - 4.54i)T \)
7 \( 1 + (-4.38 + 5.45i)T \)
good3 \( 1 + (3.43 - 3.43i)T - 9iT^{2} \)
11 \( 1 - 9.86iT - 121T^{2} \)
13 \( 1 + (8.85 + 8.85i)T + 169iT^{2} \)
17 \( 1 + (-15.3 + 15.3i)T - 289iT^{2} \)
19 \( 1 - 19.7iT - 361T^{2} \)
23 \( 1 + (-12.8 - 12.8i)T + 529iT^{2} \)
29 \( 1 - 37.0iT - 841T^{2} \)
31 \( 1 - 23.1T + 961T^{2} \)
37 \( 1 + (-23.3 - 23.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 + (50.4 + 50.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-8.81 - 8.81i)T + 2.20e3iT^{2} \)
53 \( 1 + (-12.0 + 12.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 0.298iT - 3.48e3T^{2} \)
61 \( 1 + 0.464iT - 3.72e3T^{2} \)
67 \( 1 + (11.7 - 11.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 22.8iT - 5.04e3T^{2} \)
73 \( 1 + (-60.7 - 60.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 99.5T + 6.24e3T^{2} \)
83 \( 1 + (-104. + 104. i)T - 6.88e3iT^{2} \)
89 \( 1 + 39.8T + 7.92e3T^{2} \)
97 \( 1 + (-6.68 + 6.68i)T - 9.40e3iT^{2} \)
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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

−13.71975147862289908934220955689, −12.29377978311559646509412760698, −11.43606568105310796961462606868, −10.33861868922197888499161303558, −9.715772169576477649533639780498, −7.71909891164528780791508760915, −6.79902285182525413111349713415, −5.47089027857300603629512968725, −4.78375765235564569708707068967, −3.41173333262655112798487814179, 0.862588207146058126482434577856, 2.19761013867096297038503721508, 4.72639532024103440204989201286, 5.61320522317733545088984702663, 6.41319596672986658671796376978, 8.187126934388215023753007716444, 9.416382329499225454445795711059, 10.87233844191311902391992415606, 11.75474567450261397055284389717, 12.26290706426404377496783989703

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