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Properties

Label 2-140-140.139-c1-0-15
Degree $2$
Conductor $140$
Sign $0.111 + 0.993i$
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  + (−0.331 + 1.37i)2-s − 2.13i·3-s + (−1.78 − 0.910i)4-s + (−1.94 − 1.10i)5-s + (2.93 + 0.707i)6-s + (−2.35 − 1.19i)7-s + (1.84 − 2.14i)8-s − 1.56·9-s + (2.16 − 2.30i)10-s − 2.33i·11-s + (−1.94 + 3.80i)12-s − 1.09·13-s + (2.42 − 2.84i)14-s + (−2.35 + 4.15i)15-s + (2.34 + 3.24i)16-s + 4.98·17-s + ⋯
L(s)  = 1  + (−0.234 + 0.972i)2-s − 1.23i·3-s + (−0.890 − 0.455i)4-s + (−0.869 − 0.493i)5-s + (1.19 + 0.288i)6-s + (−0.891 − 0.453i)7-s + (0.650 − 0.759i)8-s − 0.520·9-s + (0.683 − 0.729i)10-s − 0.703i·11-s + (−0.561 + 1.09i)12-s − 0.302·13-s + (0.649 − 0.760i)14-s + (−0.608 + 1.07i)15-s + (0.585 + 0.810i)16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.111 + 0.993i$
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: \((2,\ 140,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470910 - 0.421231i\)
\(L(\frac12)\) \(\approx\) \(0.470910 - 0.421231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.331 - 1.37i)T \)
5 \( 1 + (1.94 + 1.10i)T \)
7 \( 1 + (2.35 + 1.19i)T \)
good3 \( 1 + 2.13iT - 3T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
23 \( 1 - 6.04T + 23T^{2} \)
29 \( 1 - 0.561T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 9.74iT - 47T^{2} \)
53 \( 1 + 8.58iT - 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 0.620iT - 61T^{2} \)
67 \( 1 - 4.71T + 67T^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 - 9.96T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 - 3.86iT - 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−12.86881337842059207864959504410, −12.46323764391199689203404359513, −10.89082067897187859257165783740, −9.457903643628146418726591451391, −8.347086015348734524507022827288, −7.44284703896194459525808874985, −6.76002990168634668467539449535, −5.45394186989794757216075183549, −3.72801957616220722941569000900, −0.72683490191407248632891211873, 2.98181506426816068131242571592, 3.91942580182936009648197182719, 5.11598769577960765906163377403, 7.15148823093855921930890825974, 8.573170597585043681542905694115, 9.664614169815255061692444923777, 10.22267743665441494726632292261, 11.21326894312536105579687813704, 12.20552193900367650321177600306, 13.01556343673776775285994559567

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