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Properties

Label 2-140-28.19-c1-0-11
Degree $2$
Conductor $140$
Sign $0.750 + 0.660i$
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  + (1.37 − 0.334i)2-s + (−0.556 − 0.963i)3-s + (1.77 − 0.919i)4-s + (0.866 + 0.5i)5-s + (−1.08 − 1.13i)6-s + (−2.32 + 1.26i)7-s + (2.13 − 1.85i)8-s + (0.880 − 1.52i)9-s + (1.35 + 0.397i)10-s + (−1.48 + 0.856i)11-s + (−1.87 − 1.20i)12-s + 2.45i·13-s + (−2.77 + 2.51i)14-s − 1.11i·15-s + (2.30 − 3.26i)16-s + (−5.38 + 3.10i)17-s + ⋯
L(s)  = 1  + (0.971 − 0.236i)2-s + (−0.321 − 0.556i)3-s + (0.888 − 0.459i)4-s + (0.387 + 0.223i)5-s + (−0.443 − 0.464i)6-s + (−0.878 + 0.477i)7-s + (0.753 − 0.656i)8-s + (0.293 − 0.508i)9-s + (0.429 + 0.125i)10-s + (−0.447 + 0.258i)11-s + (−0.541 − 0.346i)12-s + 0.682i·13-s + (−0.740 + 0.672i)14-s − 0.287i·15-s + (0.577 − 0.816i)16-s + (−1.30 + 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.750 + 0.660i$
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: \((2,\ 140,\ (\ :1/2),\ 0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57419 - 0.593913i\)
\(L(\frac12)\) \(\approx\) \(1.57419 - 0.593913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.334i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.32 - 1.26i)T \)
good3 \( 1 + (0.556 + 0.963i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.48 - 0.856i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.45iT - 13T^{2} \)
17 \( 1 + (5.38 - 3.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.108 + 0.187i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.68 - 3.28i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
31 \( 1 + (0.0819 + 0.141i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.84 - 6.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 + 1.89iT - 43T^{2} \)
47 \( 1 + (-5.85 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.51 + 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.14 - 3.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.48 - 2.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.04iT - 71T^{2} \)
73 \( 1 + (-6.59 + 3.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.8 - 7.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.47T + 83T^{2} \)
89 \( 1 + (1.54 + 0.891i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−13.06491368463007961874295032377, −12.26963770571298501406158042781, −11.31408142328607783729508715420, −10.18048414972109575161994687432, −9.067378830745308461805304079878, −7.02502922479952433569806843958, −6.51302616069982027647233901748, −5.34156620570152557480952141018, −3.69457222908250641584644773258, −2.09601040422981231928594648026, 2.80346619993359241845071698174, 4.36102445460813411422447331352, 5.33850674727025935668412346213, 6.53684006259011250462461897867, 7.66137043026446271791803503989, 9.270661125707983396359764524117, 10.56168303288475245900651491866, 11.09648726612582890232016969069, 12.72517269236764713450807600550, 13.16817861319452515345466367725

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