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Properties

Label 2-140-140.19-c1-0-5
Degree $2$
Conductor $140$
Sign $0.927 + 0.373i$
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 + (2.07 − 0.832i)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 + (−2.04 − 2.41i)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 + (0.928 − 0.372i)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 + (−0.645 − 0.763i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.927 + 0.373i$
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: \((2,\ 140,\ (\ :1/2),\ 0.927 + 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758969 - 0.146949i\)
\(L(\frac12)\) \(\approx\) \(0.758969 - 0.146949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 + 0.832i)T \)
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

−12.58501403301359993923182026640, −11.95690267189005473696181461523, −10.89398393570554460465695016583, −10.35902768242222634019984229809, −9.279716345280124216997015788040, −8.242569384302157322426893474234, −6.07761933736819690504425859514, −5.16547332076856614207829734498, −4.00149467190943145153238234007, −1.55062708636708619017675728534, 1.37765899717681981274154298293, 4.87373651269822790760248001435, 5.64128662932123948140878134573, 6.87723928722450102214822987672, 7.32334444318669255779174669179, 9.031296969779261772466379218430, 10.12056514954209526511693950427, 11.20440763279054898794323146855, 12.15439934002735158198798589250, 13.44559013631772926317106193912

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