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Properties

Label 2-140-140.139-c1-0-8
Degree $2$
Conductor $140$
Sign $0.910 + 0.413i$
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 + (−0.874 − 3.03i)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 + (−0.276 − 0.961i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.910 + 0.413i$
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: \((2,\ 140,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28043 - 0.277346i\)
\(L(\frac12)\) \(\approx\) \(1.28043 - 0.277346i\)
\(L(\frac{3}{2})\) 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

−13.11858504653943726155799155187, −11.88514197098269191055204383442, −10.87764898525537671549700588250, −10.23364342602938280543219177879, −9.053340188376516624295720107823, −8.589120372318897719218431254504, −5.89379517535069694946361037134, −4.95493947249890316574534112561, −3.95082214716718244241299046618, −2.08474065639691169435157347492, 1.97856124390435057807884014007, 4.37483987094450817747511022240, 5.89017666218991051316235524621, 6.82377359839085400061651675108, 7.56876963662897004933914483559, 8.661540292170428244781915637149, 10.01721466094714763178737240981, 11.37128760097359032571605168977, 12.76239899315327204181053550522, 13.28742984125756306767836442527

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