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Properties

Label 2-140-28.27-c1-0-2
Degree $2$
Conductor $140$
Sign $0.978 - 0.204i$
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  + (−0.780 − 1.17i)2-s − 3.02·3-s + (−0.780 + 1.84i)4-s + i·5-s + (2.35 + 3.56i)6-s + (2.17 − 1.51i)7-s + (2.78 − 0.516i)8-s + 6.12·9-s + (1.17 − 0.780i)10-s + 4.71i·11-s + (2.35 − 5.56i)12-s + 2i·13-s + (−3.47 − 1.38i)14-s − 3.02i·15-s + (−2.78 − 2.87i)16-s + 1.12i·17-s + ⋯
L(s)  = 1  + (−0.552 − 0.833i)2-s − 1.74·3-s + (−0.390 + 0.920i)4-s + 0.447i·5-s + (0.962 + 1.45i)6-s + (0.821 − 0.570i)7-s + (0.983 − 0.182i)8-s + 2.04·9-s + (0.372 − 0.246i)10-s + 1.42i·11-s + (0.680 − 1.60i)12-s + 0.554i·13-s + (−0.929 − 0.369i)14-s − 0.779i·15-s + (−0.695 − 0.718i)16-s + 0.272i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.978 - 0.204i$
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: \((2,\ 140,\ (\ :1/2),\ 0.978 - 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491027 + 0.0508646i\)
\(L(\frac12)\) \(\approx\) \(0.491027 + 0.0508646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.780 + 1.17i)T \)
5 \( 1 - iT \)
7 \( 1 + (-2.17 + 1.51i)T \)
good3 \( 1 + 3.02T + 3T^{2} \)
11 \( 1 - 4.71iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 1.12iT - 17T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 + 0.371iT - 43T^{2} \)
47 \( 1 + 5.08T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 2.06T + 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 3.76iT - 67T^{2} \)
71 \( 1 + 7.36iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 1.32iT - 79T^{2} \)
83 \( 1 + 3.02T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 1.12iT - 97T^{2} \)
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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

−12.71260399283989724298471004918, −11.70812693556145738720258878849, −11.35373350881951737283864932598, −10.31170621364235501155034576614, −9.631521527079264100959728033740, −7.64895512927598190751912184772, −6.92234384495071455332750676106, −5.18409885318345428302534497699, −4.14417934128257587207001844612, −1.54326506653660066469279922125, 0.851456247322789984511265991302, 4.78406598656606920428512972504, 5.54514950020594540435832478849, 6.32179755233776608514433828565, 7.76080088807788688158615048810, 8.809776468074845424281832661366, 10.17061094618239068625188660524, 11.12453697312912148277507416745, 11.79839139500756639322030831472, 13.05388972961655883440613648715

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