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Properties

Label 2-140-140.27-c2-0-0
Degree $2$
Conductor $140$
Sign $-0.189 + 0.981i$
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  + (−0.427 + 1.95i)2-s + (−2.73 + 2.73i)3-s + (−3.63 − 1.67i)4-s + (−2.65 + 4.23i)5-s + (−4.17 − 6.51i)6-s + (1.71 + 6.78i)7-s + (4.81 − 6.38i)8-s − 5.96i·9-s + (−7.14 − 6.99i)10-s − 14.7i·11-s + (14.5 − 5.36i)12-s + (−2.13 − 2.13i)13-s + (−13.9 + 0.439i)14-s + (−4.32 − 18.8i)15-s + (10.4 + 12.1i)16-s + (−7.25 + 7.25i)17-s + ⋯
L(s)  = 1  + (−0.213 + 0.976i)2-s + (−0.911 + 0.911i)3-s + (−0.908 − 0.417i)4-s + (−0.530 + 0.847i)5-s + (−0.695 − 1.08i)6-s + (0.244 + 0.969i)7-s + (0.602 − 0.798i)8-s − 0.662i·9-s + (−0.714 − 0.699i)10-s − 1.34i·11-s + (1.20 − 0.447i)12-s + (−0.164 − 0.164i)13-s + (−0.999 + 0.0313i)14-s + (−0.288 − 1.25i)15-s + (0.650 + 0.759i)16-s + (−0.426 + 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.189 + 0.981i$
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: \((2,\ 140,\ (\ :1),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.237505 - 0.287622i\)
\(L(\frac12)\) \(\approx\) \(0.237505 - 0.287622i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.427 - 1.95i)T \)
5 \( 1 + (2.65 - 4.23i)T \)
7 \( 1 + (-1.71 - 6.78i)T \)
good3 \( 1 + (2.73 - 2.73i)T - 9iT^{2} \)
11 \( 1 + 14.7iT - 121T^{2} \)
13 \( 1 + (2.13 + 2.13i)T + 169iT^{2} \)
17 \( 1 + (7.25 - 7.25i)T - 289iT^{2} \)
19 \( 1 + 8.82iT - 361T^{2} \)
23 \( 1 + (-19.2 - 19.2i)T + 529iT^{2} \)
29 \( 1 - 49.9iT - 841T^{2} \)
31 \( 1 + 48.9T + 961T^{2} \)
37 \( 1 + (37.2 + 37.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 10.8iT - 1.68e3T^{2} \)
43 \( 1 + (3.78 + 3.78i)T + 1.84e3iT^{2} \)
47 \( 1 + (-28.1 - 28.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (15.9 - 15.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 12.0iT - 3.48e3T^{2} \)
61 \( 1 + 71.8iT - 3.72e3T^{2} \)
67 \( 1 + (7.58 - 7.58i)T - 4.48e3iT^{2} \)
71 \( 1 - 61.1iT - 5.04e3T^{2} \)
73 \( 1 + (-53.2 - 53.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 77.9T + 6.24e3T^{2} \)
83 \( 1 + (82.1 - 82.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 40.2T + 7.92e3T^{2} \)
97 \( 1 + (63.6 - 63.6i)T - 9.40e3iT^{2} \)
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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

−14.04014372171250307867454365506, −12.59905170884166955551406072149, −11.03424045437140463895106136557, −10.92926680583321611754562806645, −9.382264002531357858970545184533, −8.449896674093292586876221454111, −7.08613212588649177287075530981, −5.85382588173633679809053483231, −5.13316375734960884229207564428, −3.57228674826203671243428554714, 0.30538836943020198068587373674, 1.68369895979713934397631999073, 4.12144215452634951271944734148, 5.08697389611428566562271655394, 7.00441639027426116472935372555, 7.82924724653812512121082122521, 9.204877106603897288429909614336, 10.38027615422901699855738876896, 11.45032420584305653749082675820, 12.14003506827840041238931385939

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