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Properties

Label 2-140-140.139-c1-0-1
Degree $2$
Conductor $140$
Sign $-0.597 - 0.801i$
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  − 1.41·2-s + 3.16i·3-s + 2.00·4-s + 2.23i·5-s − 4.47i·6-s + (2.12 − 1.58i)7-s − 2.82·8-s − 7.00·9-s − 3.16i·10-s + 6.32i·12-s + (−3 + 2.23i)14-s − 7.07·15-s + 4.00·16-s + 9.89·18-s + 4.47i·20-s + (5.00 + 6.70i)21-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.82i·3-s + 1.00·4-s + 0.999i·5-s − 1.82i·6-s + (0.801 − 0.597i)7-s − 1.00·8-s − 2.33·9-s − 1.00i·10-s + 1.82i·12-s + (−0.801 + 0.597i)14-s − 1.82·15-s + 1.00·16-s + 2.33·18-s + 1.00i·20-s + (1.09 + 1.46i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.597 - 0.801i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.597 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327974 + 0.653512i\)
\(L(\frac12)\) \(\approx\) \(0.327974 + 0.653512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 - 2.23iT \)
7 \( 1 + (-2.12 + 1.58i)T \)
good3 \( 1 - 3.16iT - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−14.10586633530105606298777441347, −11.80761524707056388647112783227, −10.87505614780556068669936945540, −10.53283816079493452676757306396, −9.604658773732696307506853409906, −8.553375213129956667900656903812, −7.35944713979465337270479120165, −5.84987879027353342347710013322, −4.27960996874906013470724004554, −2.89485568630563151867614750405, 1.11334332249999428311661756511, 2.34709449904617773783487411458, 5.42626421453366896310194279543, 6.55806189615336503593621864330, 7.78700650469074565165927284281, 8.347509671280928183043050335475, 9.233569466333968347394545005428, 11.03066874588294619886744333876, 12.01870468129063764929823642111, 12.43199659394943402428795624940

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