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Properties

Label 2-140-35.17-c1-0-1
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

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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: $\chi_{140} (17, \cdot )$
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 \( 1 + (-11.1 + 11.1i)T - 97iT^{2} \)
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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

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

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