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Properties

Label 2-140-28.27-c1-0-15
Degree $2$
Conductor $140$
Sign $-0.981 + 0.188i$
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  + (0.366 − 1.36i)2-s − 1.73·3-s + (−1.73 − i)4-s − i·5-s + (−0.633 + 2.36i)6-s + (−1.73 − 2i)7-s + (−2 + 1.99i)8-s + (−1.36 − 0.366i)10-s − 0.267i·11-s + (2.99 + 1.73i)12-s − 0.464i·13-s + (−3.36 + 1.63i)14-s + 1.73i·15-s + (1.99 + 3.46i)16-s − 6.46i·17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s − 1.00·3-s + (−0.866 − 0.5i)4-s − 0.447i·5-s + (−0.258 + 0.965i)6-s + (−0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (−0.431 − 0.115i)10-s − 0.0807i·11-s + (0.866 + 0.499i)12-s − 0.128i·13-s + (−0.899 + 0.436i)14-s + 0.447i·15-s + (0.499 + 0.866i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.981 + 0.188i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.981 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0589062 - 0.617788i\)
\(L(\frac12)\) \(\approx\) \(0.0589062 - 0.617788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + iT \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + 1.73T + 3T^{2} \)
11 \( 1 + 0.267iT - 11T^{2} \)
13 \( 1 + 0.464iT - 13T^{2} \)
17 \( 1 + 6.46iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 9.46T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 9.46iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 + 13.3iT - 97T^{2} \)
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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

−12.35286703727573479094863423883, −11.79678084710694179259016051729, −10.77788176836264166678661290052, −9.917001754986770221064083038267, −8.868877961939504159867187409771, −7.11378647964160262014460832949, −5.65956700888107471052541921644, −4.75297579295402906626388701752, −3.17950822102262844129885461979, −0.66756306154638102591564443936, 3.39919770862336868549693868751, 5.16220767501999316703122289698, 6.02985952146808533006216944611, 6.81842044350380925908750047663, 8.228199031050160078963874641245, 9.404360551564499699705224351273, 10.58583817224558325641605871202, 11.91930144487131302115575156422, 12.52931094571205585742381997523, 13.72871571461879644589810260012

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