↑

Properties

Label 2-140-35.3-c1-0-3
Degree $2$
Conductor $140$
Sign $-0.288 + 0.957i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.837 − 3.12i)3-s + (2.23 + 0.121i)5-s + (−2.49 − 0.870i)7-s + (−6.46 + 3.73i)9-s + (0.615 − 1.06i)11-s + (2.44 − 2.44i)13-s + (−1.48 − 7.07i)15-s + (1.47 − 0.395i)17-s + (2.14 + 3.71i)19-s + (−0.626 + 8.53i)21-s + (−0.267 + 0.999i)23-s + (4.97 + 0.544i)25-s + (10.2 + 10.2i)27-s − 3.02i·29-s + (4.67 + 2.70i)31-s + ⋯
L(s)  = 1  + (−0.483 − 1.80i)3-s + (0.998 + 0.0545i)5-s + (−0.944 − 0.328i)7-s + (−2.15 + 1.24i)9-s + (0.185 − 0.321i)11-s + (0.677 − 0.677i)13-s + (−0.384 − 1.82i)15-s + (0.358 − 0.0960i)17-s + (0.492 + 0.853i)19-s + (−0.136 + 1.86i)21-s + (−0.0558 + 0.208i)23-s + (0.994 + 0.108i)25-s + (1.96 + 1.96i)27-s − 0.562i·29-s + (0.840 + 0.485i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.288 + 0.957i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591190 - 0.795791i\)
\(L(\frac12)\) \(\approx\) \(0.591190 - 0.795791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.23 - 0.121i)T \)
7 \( 1 + (2.49 + 0.870i)T \)
good3 \( 1 + (0.837 + 3.12i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.615 + 1.06i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 13iT^{2} \)
17 \( 1 + (-1.47 + 0.395i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.14 - 3.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.267 - 0.999i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.02iT - 29T^{2} \)
31 \( 1 + (-4.67 - 2.70i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.22 + 0.328i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.26iT - 41T^{2} \)
43 \( 1 + (-6.08 - 6.08i)T + 43iT^{2} \)
47 \( 1 + (-1.43 + 5.36i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.0 - 3.21i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.71 + 2.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.82 - 2.78i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.08 - 4.04i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (-1.36 - 5.07i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.35 - 2.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.84 - 8.84i)T - 83iT^{2} \)
89 \( 1 + (3.05 + 5.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.91412846221452989107656896238, −12.18580030858823951117223862657, −10.96743578301392553571672344896, −9.843620227159808310479339366012, −8.412333601839008383958123905503, −7.27995621408033981793120753245, −6.22964191470780402916230074150, −5.71053039356318825204936140623, −2.94436382440885071260627320325, −1.21951076676920343975389493176, 3.07438739292492255360784098916, 4.47393088943720852302477891255, 5.64304147574613165577196697862, 6.50697706393736222245990209094, 8.923210152896858581041908771569, 9.446850611401809791277598535855, 10.22683146066414372401887753652, 11.15068991949306786261308860056, 12.29508117826034878608185984356, 13.62031956318436413697758691772

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