↑

Properties

Label 2-140-35.33-c1-0-0
Degree $2$
Conductor $140$
Sign $0.389 - 0.921i$
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  + (−3.12 − 0.837i)3-s + (1.01 + 1.99i)5-s + (0.870 + 2.49i)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 + (0.395 − 1.47i)17-s + (−2.14 + 3.71i)19-s + (−0.626 − 8.53i)21-s + (0.999 − 0.267i)23-s + (−2.95 + 4.03i)25-s + (−10.2 − 10.2i)27-s − 3.02i·29-s + (4.67 − 2.70i)31-s + ⋯
L(s)  = 1  + (−1.80 − 0.483i)3-s + (0.452 + 0.892i)5-s + (0.328 + 0.944i)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 + (0.0960 − 0.358i)17-s + (−0.492 + 0.853i)19-s + (−0.136 − 1.86i)21-s + (0.208 − 0.0558i)23-s + (−0.591 + 0.806i)25-s + (−1.96 − 1.96i)27-s − 0.562i·29-s + (0.840 − 0.485i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.389 - 0.921i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.389 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539992 + 0.358050i\)
\(L(\frac12)\) \(\approx\) \(0.539992 + 0.358050i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.01 - 1.99i)T \)
7 \( 1 + (-0.870 - 2.49i)T \)
good3 \( 1 + (3.12 + 0.837i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.615 - 1.06i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (-0.395 + 1.47i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.14 - 3.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.999 + 0.267i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.02iT - 29T^{2} \)
31 \( 1 + (-4.67 + 2.70i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.328 - 1.22i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.26iT - 41T^{2} \)
43 \( 1 + (-6.08 - 6.08i)T + 43iT^{2} \)
47 \( 1 + (-5.36 + 1.43i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.21 + 12.0i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (4.04 + 1.08i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (-5.07 - 1.36i)T + (63.2 + 36.5i)T^{2} \)
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

−13.04502993715146928025293324311, −12.00968429167243450346729701548, −11.55731389436805773616767458462, −10.50801248338225534162783407020, −9.575893680526473810291875080435, −7.65431445297840182829213068480, −6.55557597169277550532020519806, −5.86101631008835839264147049627, −4.69368239203309818440174139807, −2.06635450120563584829682270020, 0.860537810009495429094368963438, 4.28807609684337446176690496324, 5.10857076975769081523781786454, 6.12483718364897075003343414168, 7.36405270115491398756017211830, 9.055290431908148190923377838065, 10.30405605696663182443789198232, 10.77028745733891122991440700086, 11.97458287508504939782684400161, 12.68703770473068183160120744460

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