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Properties

Label 2-140-28.19-c1-0-10
Degree $2$
Conductor $140$
Sign $0.727 + 0.686i$
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  + (1.05 − 0.942i)2-s + (0.450 + 0.780i)3-s + (0.224 − 1.98i)4-s + (−0.866 − 0.5i)5-s + (1.21 + 0.398i)6-s + (2.29 + 1.30i)7-s + (−1.63 − 2.30i)8-s + (1.09 − 1.89i)9-s + (−1.38 + 0.288i)10-s + (−3.24 + 1.87i)11-s + (1.65 − 0.720i)12-s + 2.41i·13-s + (3.65 − 0.786i)14-s − 0.901i·15-s + (−3.89 − 0.893i)16-s + (−0.505 + 0.291i)17-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)2-s + (0.260 + 0.450i)3-s + (0.112 − 0.993i)4-s + (−0.387 − 0.223i)5-s + (0.494 + 0.162i)6-s + (0.869 + 0.494i)7-s + (−0.578 − 0.815i)8-s + (0.364 − 0.631i)9-s + (−0.437 + 0.0912i)10-s + (−0.977 + 0.564i)11-s + (0.477 − 0.207i)12-s + 0.671i·13-s + (0.977 − 0.210i)14-s − 0.232i·15-s + (−0.974 − 0.223i)16-s + (−0.122 + 0.0707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51627 - 0.602703i\)
\(L(\frac12)\) \(\approx\) \(1.51627 - 0.602703i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 + 0.942i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.29 - 1.30i)T \)
good3 \( 1 + (-0.450 - 0.780i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.24 - 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.41iT - 13T^{2} \)
17 \( 1 + (0.505 - 0.291i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.07 - 5.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.73 + 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.435T + 29T^{2} \)
31 \( 1 + (1.26 + 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.65 + 9.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.35iT - 41T^{2} \)
43 \( 1 - 5.80iT - 43T^{2} \)
47 \( 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.55 - 2.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.73 + 3.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.99 + 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.52 + 4.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.96iT - 71T^{2} \)
73 \( 1 + (-8.48 + 4.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.397 + 0.229i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + (8.55 + 4.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.54iT - 97T^{2} \)
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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

−12.74760199313875191910957336817, −12.19426007219403964968425835023, −11.11907057057630546404580334982, −10.13228689678029649169606485277, −9.112660710245761400245143288739, −7.82180742586486276777287909524, −6.14727836368313211976366631084, −4.78218277491513923428527885602, −3.92058722558684412646066651561, −2.11382729163852091343236472100, 2.65107506927194138372006946687, 4.33655550833731760856653921653, 5.43671977429896200813599271191, 7.00413715997886162678685351831, 7.82309515260218287084229605712, 8.471211872099136082982058530948, 10.55304466097550970140774310331, 11.34022943142358581334421184072, 12.63590317335905382777674322269, 13.47513048592132123555270679449

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