Properties

Label 2-140-140.19-c1-0-3
Degree $2$
Conductor $140$
Sign $-0.540 - 0.841i$
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  + (0.999 + 1.00i)2-s + (−1.28 + 0.739i)3-s + (−0.00125 + 1.99i)4-s + (−2.22 − 0.175i)5-s + (−2.02 − 0.542i)6-s + (0.664 + 2.56i)7-s + (−2.00 + 1.99i)8-s + (−0.406 + 0.703i)9-s + (−2.05 − 2.40i)10-s + (5.32 − 3.07i)11-s + (−1.47 − 2.56i)12-s + 3.33·13-s + (−1.89 + 3.22i)14-s + (2.98 − 1.42i)15-s + (−3.99 − 0.00500i)16-s + (−1.27 − 2.20i)17-s + ⋯
L(s)  = 1  + (0.706 + 0.707i)2-s + (−0.739 + 0.426i)3-s + (−0.000625 + 0.999i)4-s + (−0.996 − 0.0784i)5-s + (−0.824 − 0.221i)6-s + (0.250 + 0.967i)7-s + (−0.707 + 0.706i)8-s + (−0.135 + 0.234i)9-s + (−0.649 − 0.760i)10-s + (1.60 − 0.927i)11-s + (−0.426 − 0.739i)12-s + 0.924·13-s + (−0.507 + 0.861i)14-s + (0.770 − 0.367i)15-s + (−0.999 − 0.00125i)16-s + (−0.309 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529901 + 0.970671i\)
\(L(\frac12)\) \(\approx\) \(0.529901 + 0.970671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.999 - 1.00i)T \)
5 \( 1 + (2.22 + 0.175i)T \)
7 \( 1 + (-0.664 - 2.56i)T \)
good3 \( 1 + (1.28 - 0.739i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.32 + 3.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.33T + 13T^{2} \)
17 \( 1 + (1.27 + 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.352 - 0.611i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.983 - 1.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.17T + 29T^{2} \)
31 \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.90 + 3.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.53iT - 41T^{2} \)
43 \( 1 + 4.59T + 43T^{2} \)
47 \( 1 + (-3.78 - 2.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.80 + 2.77i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.40 + 5.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.07 - 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.45 + 2.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.37iT - 71T^{2} \)
73 \( 1 + (-1.27 - 2.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.38 + 3.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.70iT - 83T^{2} \)
89 \( 1 + (5.19 + 3.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.46T + 97T^{2} \)
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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

−13.72845822280726269165973363397, −12.16705784220136188446073961274, −11.72513589913058713699254808521, −10.97995847148146372129067143214, −8.877331963652690314011207786642, −8.326873258532188815897700688607, −6.72156884605318144591157748456, −5.76116444529453436312070258245, −4.62955530815253427175967016118, −3.42685781949007400553538904067, 1.14860003551690990327430678587, 3.72776887761642013953239020040, 4.49909304087241203715619430438, 6.31781094376780091577778636683, 6.97873312987589072574194109799, 8.694690326512961162895985337947, 10.16995510151508968773123033561, 11.21548024054284294963093754745, 11.79319365323003359432680919938, 12.50980006104491236850796599586

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