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Properties

Label 2-140-140.39-c0-0-1
Degree $2$
Conductor $140$
Sign $-0.0633 + 0.997i$
Analytic cond. $0.0698691$
Root an. cond. $0.264327$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + 0.999·20-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + 0.999·20-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(0.0698691\)
Root analytic conductor: \(0.264327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5142261617\)
\(L(\frac12)\) \(\approx\) \(0.5142261617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−13.02974905280989370564192259687, −12.17860024544497028843624594850, −11.42040436361054528192267196081, −9.877769206049269076046763875976, −8.868601555853835613352467318640, −8.180747780401275503283824113235, −7.10387591591902298203109404153, −5.13376199973804018896584366820, −3.41763793665203501307430262132, −1.84798733176887999444039248112, 3.43782950479146480647989201129, 4.57530209214505313204504732398, 6.41661967871024913557960236053, 7.26104126294525014046992686963, 8.416143973274795439126093561837, 9.614591293890291152644650722770, 10.27748392754102519390547451999, 11.16716525752643526132281201912, 12.96999685799762280518674606496, 14.17587321847294524728955915446

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