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Properties

Label 2-140-28.19-c1-0-6
Degree $2$
Conductor $140$
Sign $0.420 - 0.907i$
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  + (1.17 + 0.790i)2-s + (0.331 + 0.573i)3-s + (0.750 + 1.85i)4-s + (−0.866 − 0.5i)5-s + (−0.0650 + 0.934i)6-s + (−1.68 + 2.03i)7-s + (−0.585 + 2.76i)8-s + (1.28 − 2.21i)9-s + (−0.620 − 1.27i)10-s + (3.12 − 1.80i)11-s + (−0.815 + 1.04i)12-s − 5.83i·13-s + (−3.58 + 1.05i)14-s − 0.662i·15-s + (−2.87 + 2.78i)16-s + (−1.18 + 0.684i)17-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)2-s + (0.191 + 0.331i)3-s + (0.375 + 0.926i)4-s + (−0.387 − 0.223i)5-s + (−0.0265 + 0.381i)6-s + (−0.637 + 0.770i)7-s + (−0.207 + 0.978i)8-s + (0.426 − 0.739i)9-s + (−0.196 − 0.401i)10-s + (0.943 − 0.544i)11-s + (−0.235 + 0.301i)12-s − 1.61i·13-s + (−0.959 + 0.282i)14-s − 0.171i·15-s + (−0.718 + 0.695i)16-s + (−0.287 + 0.165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.420 - 0.907i$
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: \((2,\ 140,\ (\ :1/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37793 + 0.880217i\)
\(L(\frac12)\) \(\approx\) \(1.37793 + 0.880217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.790i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.68 - 2.03i)T \)
good3 \( 1 + (-0.331 - 0.573i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.83iT - 13T^{2} \)
17 \( 1 + (1.18 - 0.684i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 - 3.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.81 + 1.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + (4.43 + 7.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.39 - 9.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.832iT - 41T^{2} \)
43 \( 1 + 3.10iT - 43T^{2} \)
47 \( 1 + (3.44 - 5.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.70 - 6.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 0.742i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.19 + 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.52iT - 71T^{2} \)
73 \( 1 + (4.47 - 2.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.82 + 5.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 + (-8.13 - 4.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.343iT - 97T^{2} \)
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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

−13.24932211768821681259298386430, −12.43998488638207539729203367155, −11.78515791221827796010212245638, −10.25889106038093204037728280982, −8.924978994076872519456040010802, −8.081218128921680238673867430097, −6.55651304358454672462881835108, −5.71828565087724300844219882293, −4.12308129763299632577809290430, −3.12891666770610014726860555308, 1.93273955252425874644930396731, 3.76308859536118095075199049963, 4.68280760598343557955184480384, 6.73336227221287755837503504821, 7.04679604460011224438283217014, 9.032504040647008296332686249604, 10.14368329437689123706764561877, 11.12187187313054173119134152168, 12.07688888508455045876332313578, 13.00206415452976823921215845311

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