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Properties

Label 2-140-28.27-c1-0-4
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  + (−1.36 − 0.366i)2-s + 1.73·3-s + (1.73 + i)4-s + i·5-s + (−2.36 − 0.633i)6-s + (1.73 + 2i)7-s + (−1.99 − 2i)8-s + (0.366 − 1.36i)10-s + 3.73i·11-s + (2.99 + 1.73i)12-s − 6.46i·13-s + (−1.63 − 3.36i)14-s + 1.73i·15-s + (1.99 + 3.46i)16-s − 0.464i·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + 1.00·3-s + (0.866 + 0.5i)4-s + 0.447i·5-s + (−0.965 − 0.258i)6-s + (0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (0.115 − 0.431i)10-s + 1.12i·11-s + (0.866 + 0.499i)12-s − 1.79i·13-s + (−0.436 − 0.899i)14-s + 0.447i·15-s + (0.499 + 0.866i)16-s − 0.112i·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

\(L(1)\) \(\approx\) \(1.00243 + 0.0955819i\)
\(L(\frac12)\) \(\approx\) \(1.00243 + 0.0955819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
5 \( 1 - iT \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 - 3.73iT - 11T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 + 0.464iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
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 \( 1 + 2.53iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 - 0.928iT - 73T^{2} \)
79 \( 1 + 2.66iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−13.01279461450136737030590057380, −12.06913601291104380686308904301, −10.96751445537355976307273780708, −9.925140027950758437894106989988, −9.002827187948285875855316348389, −8.009373213722982671850439819836, −7.34476711637577239965022710427, −5.54146860361022471232453573477, −3.27145977687743645557013697797, −2.20389589143618221305775567119, 1.67368025402566070603883848501, 3.59889789409010164205295722526, 5.51683083249260160026979033807, 7.14203736574820061046078816135, 8.000783233667318446666613834148, 8.969596023014252945070630651690, 9.559794286102011012517227700707, 11.13910621390463233819418737717, 11.65611596020320565856537845920, 13.63632025520763565419953490032

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