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Properties

Label 2-140-140.19-c1-0-9
Degree $2$
Conductor $140$
Sign $0.582 - 0.812i$
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  + (0.444 + 1.34i)2-s + (2.24 − 1.29i)3-s + (−1.60 + 1.19i)4-s + (0.316 + 2.21i)5-s + (2.74 + 2.44i)6-s + (−2.57 − 0.603i)7-s + (−2.31 − 1.62i)8-s + (1.87 − 3.23i)9-s + (−2.83 + 1.40i)10-s + (3.12 − 1.80i)11-s + (−2.05 + 4.76i)12-s + 0.818·13-s + (−0.335 − 3.72i)14-s + (3.58 + 4.56i)15-s + (1.14 − 3.83i)16-s + (−3.69 − 6.40i)17-s + ⋯
L(s)  = 1  + (0.314 + 0.949i)2-s + (1.29 − 0.749i)3-s + (−0.801 + 0.597i)4-s + (0.141 + 0.989i)5-s + (1.11 + 0.996i)6-s + (−0.973 − 0.228i)7-s + (−0.819 − 0.573i)8-s + (0.623 − 1.07i)9-s + (−0.895 + 0.445i)10-s + (0.940 − 0.543i)11-s + (−0.593 + 1.37i)12-s + 0.226·13-s + (−0.0896 − 0.995i)14-s + (0.925 + 1.17i)15-s + (0.286 − 0.958i)16-s + (−0.896 − 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.582 - 0.812i$
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: \((2,\ 140,\ (\ :1/2),\ 0.582 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41467 + 0.726406i\)
\(L(\frac12)\) \(\approx\) \(1.41467 + 0.726406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.444 - 1.34i)T \)
5 \( 1 + (-0.316 - 2.21i)T \)
7 \( 1 + (2.57 + 0.603i)T \)
good3 \( 1 + (-2.24 + 1.29i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.818T + 13T^{2} \)
17 \( 1 + (3.69 + 6.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.65 - 2.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.26 - 2.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.04T + 29T^{2} \)
31 \( 1 + (0.955 + 1.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.20 - 3.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.65iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 + (1.15 + 0.667i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.56 - 0.905i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.955 - 1.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.46 + 4.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.63 - 8.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.38iT - 71T^{2} \)
73 \( 1 + (-3.69 - 6.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.70 - 3.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + (9.19 + 5.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.32T + 97T^{2} \)
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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

−13.68528287464902437663416517284, −12.88639552306428884742858011462, −11.51423953064906071479362540264, −9.697816462653533604777537464506, −8.987211964776626325722037097082, −7.75141871272447289320830350875, −6.90180154464649025897816202776, −6.16720470765540079477310107531, −3.83215205699065139704238420291, −2.83941043708870941861757288180, 2.14794417117424492402810858918, 3.72727920269057298885082819458, 4.44838500949583319750038911817, 6.18318110012099591426938087639, 8.429463621439765391188246670124, 9.125394108600669186316107853811, 9.653755378405485135637500054910, 10.79418421316187840800247891198, 12.32600361431393614244477514860, 12.97222962334647443289751833768

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