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Properties

Label 2-140-140.27-c2-0-1
Degree $2$
Conductor $140$
Sign $-0.466 - 0.884i$
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  + (1.34 − 1.47i)2-s + (−2.39 + 2.39i)3-s + (−0.373 − 3.98i)4-s + (−4.78 + 1.46i)5-s + (0.316 + 6.76i)6-s + (−3.10 + 6.27i)7-s + (−6.39 − 4.81i)8-s − 2.46i·9-s + (−4.27 + 9.03i)10-s + 13.1i·11-s + (10.4 + 8.64i)12-s + (−11.6 − 11.6i)13-s + (5.10 + 13.0i)14-s + (7.94 − 14.9i)15-s + (−15.7 + 2.97i)16-s + (4.14 − 4.14i)17-s + ⋯
L(s)  = 1  + (0.673 − 0.739i)2-s + (−0.798 + 0.798i)3-s + (−0.0933 − 0.995i)4-s + (−0.956 + 0.292i)5-s + (0.0527 + 1.12i)6-s + (−0.443 + 0.896i)7-s + (−0.798 − 0.601i)8-s − 0.273i·9-s + (−0.427 + 0.903i)10-s + 1.19i·11-s + (0.869 + 0.720i)12-s + (−0.893 − 0.893i)13-s + (0.364 + 0.931i)14-s + (0.529 − 0.996i)15-s + (−0.982 + 0.185i)16-s + (0.243 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.466 - 0.884i$
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: \((2,\ 140,\ (\ :1),\ -0.466 - 0.884i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.281908 + 0.467098i\)
\(L(\frac12)\) \(\approx\) \(0.281908 + 0.467098i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 1.47i)T \)
5 \( 1 + (4.78 - 1.46i)T \)
7 \( 1 + (3.10 - 6.27i)T \)
good3 \( 1 + (2.39 - 2.39i)T - 9iT^{2} \)
11 \( 1 - 13.1iT - 121T^{2} \)
13 \( 1 + (11.6 + 11.6i)T + 169iT^{2} \)
17 \( 1 + (-4.14 + 4.14i)T - 289iT^{2} \)
19 \( 1 - 23.6iT - 361T^{2} \)
23 \( 1 + (15.5 + 15.5i)T + 529iT^{2} \)
29 \( 1 + 11.8iT - 841T^{2} \)
31 \( 1 - 26.5T + 961T^{2} \)
37 \( 1 + (-33.9 - 33.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 37.7iT - 1.68e3T^{2} \)
43 \( 1 + (8.31 + 8.31i)T + 1.84e3iT^{2} \)
47 \( 1 + (16.9 + 16.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (62.1 - 62.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 12.5iT - 3.48e3T^{2} \)
61 \( 1 + 38.8iT - 3.72e3T^{2} \)
67 \( 1 + (42.2 - 42.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 44.7iT - 5.04e3T^{2} \)
73 \( 1 + (-99.8 - 99.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 10.1T + 6.24e3T^{2} \)
83 \( 1 + (89.2 - 89.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 3.55T + 7.92e3T^{2} \)
97 \( 1 + (-54.3 + 54.3i)T - 9.40e3iT^{2} \)
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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

−12.71781582434439832776688816051, −12.14870919029056693115242618326, −11.44838669593174299881161270733, −10.14996469462210431237556926110, −9.860092986034965643086688067508, −7.969065566364307494108515061248, −6.31350878874737528155073209685, −5.13556077392646308921308870549, −4.23065739482683086003966168851, −2.71863175203560014923107158557, 0.30879458538465443090051946923, 3.48934417762428095438291390326, 4.74562767377754873539482447426, 6.16841091126959155998766404257, 7.04824418466504162651363324416, 7.81805986682868240091705430125, 9.178219557062963288782750914063, 11.13057929037426957792047706476, 11.77943032645194255082207585127, 12.66359115354733445526417535713

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