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Properties

Label 2-140-28.3-c1-0-1
Degree $2$
Conductor $140$
Sign $-0.869 - 0.493i$
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  + (0.288 + 1.38i)2-s + (−0.450 + 0.780i)3-s + (−1.83 + 0.798i)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 + (−0.942 − 1.05i)10-s + (3.24 + 1.87i)11-s + (0.202 − 1.79i)12-s − 2.41i·13-s + (−2.47 − 2.80i)14-s − 0.901i·15-s + (2.72 − 2.92i)16-s + (−0.505 − 0.291i)17-s + ⋯
L(s)  = 1  + (0.204 + 0.978i)2-s + (−0.260 + 0.450i)3-s + (−0.916 + 0.399i)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 + (−0.297 − 0.333i)10-s + (0.977 + 0.564i)11-s + (0.0585 − 0.517i)12-s − 0.671i·13-s + (−0.661 − 0.750i)14-s − 0.232i·15-s + (0.680 − 0.732i)16-s + (−0.122 − 0.0707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226799 + 0.859735i\)
\(L(\frac12)\) \(\approx\) \(0.226799 + 0.859735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.288 - 1.38i)T \)
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

−13.65905593524845528156400474899, −12.69632156709578770162406590251, −11.78892540843155119846605578171, −10.21066742383786144352322939652, −9.454045438128031468577129329512, −8.161743505233273054059105928874, −7.05195146919997810955339991380, −5.97414710420107783454471262492, −4.73413186132773710673774891831, −3.45523543922301035904597343651, 0.976893067246798567731117961074, 3.28911015247854392265652469236, 4.39168913388479589426163558759, 6.13137853956441586446388974419, 7.21320642619173869027643058665, 9.041022766883913267403762658382, 9.540036359349927230262300581783, 11.05055019926007563765964935205, 11.72469491380793759638189608323, 12.68224005596429743190599568694

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