Properties

Label 2-140-140.19-c1-0-7
Degree $2$
Conductor $140$
Sign $0.754 - 0.656i$
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  + (1.38 − 0.285i)2-s + (−2.24 + 1.29i)3-s + (1.83 − 0.791i)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 + (1.07 + 2.97i)10-s + (−3.12 + 1.80i)11-s + (−3.10 + 4.16i)12-s + 0.818·13-s + (3.74 + 0.100i)14-s + (−3.58 − 4.56i)15-s + (2.74 − 2.90i)16-s + (−3.69 − 6.40i)17-s + ⋯
L(s)  = 1  + (0.979 − 0.202i)2-s + (−1.29 + 0.749i)3-s + (0.918 − 0.395i)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 + (0.338 + 0.940i)10-s + (−0.940 + 0.543i)11-s + (−0.895 + 1.20i)12-s + 0.226·13-s + (0.999 + 0.0267i)14-s + (−0.925 − 1.17i)15-s + (0.686 − 0.727i)16-s + (−0.896 − 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.754 - 0.656i$
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: \((2,\ 140,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32075 + 0.493811i\)
\(L(\frac12)\) \(\approx\) \(1.32075 + 0.493811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.285i)T \)
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

−13.29001052166906689634195579561, −11.91318714156635959306698998438, −11.21947629853907650270390439447, −10.76315281118795210193021981707, −9.702784955610301432719359730589, −7.48514070044377907630256786648, −6.41231850699030807444154876392, −5.20130906468401458765536155179, −4.58406768420736956779686311488, −2.64893037172454810931576313578, 1.63020391908598930536916259543, 4.30149122619109876344597308801, 5.44922922159791721719164505803, 6.01886004677656426419441339853, 7.50686139896238858333451538233, 8.390010382814742420877407079452, 10.62079364115763465901460306065, 11.30023230395489680091227393623, 12.19935680978896623692850719365, 13.02997817986658145921521892361

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