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Properties

Label 2-140-28.27-c1-0-10
Degree $2$
Conductor $140$
Sign $0.533 + 0.846i$
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 + (−0.0846 + 3.74i)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 + (−0.0226 + 0.999i)14-s − 0.779i·15-s + (−0.695 − 0.718i)16-s − 0.272i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.533 + 0.846i$
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: \((2,\ 140,\ (\ :1/2),\ 0.533 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11280 - 0.614116i\)
\(L(\frac12)\) \(\approx\) \(1.11280 - 0.614116i\)
\(L(\frac{3}{2})\) 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

−13.04290498334033982964585768275, −12.31544476343223725580702206096, −10.54500415675941571566620423349, −9.626074271076657112406058609528, −9.139592063103008043468949050748, −7.901075500195397599472843459131, −7.20145555232862999180074225308, −4.37313079396885604715450589352, −3.34465993432659366823520021504, −1.96614805819024225451468983340, 2.46967958064662831402488955822, 3.92712216470713048158606206356, 6.05219150961933518203239323790, 7.05364621227989553546874538705, 8.411950183698144766150102366059, 8.784613871405256071282821594072, 9.784584130215859156714788772431, 10.86106249204365146763988790644, 12.81133634291557327099319875301, 13.69432147460792029584269969780

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