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Properties

Label 2-140-35.17-c1-0-2
Degree $2$
Conductor $140$
Sign $0.962 + 0.271i$
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  + (1.52 − 0.409i)3-s + (2.14 − 0.632i)5-s + (−2.59 − 0.512i)7-s + (−0.434 + 0.250i)9-s + (−0.342 + 0.593i)11-s + (1.04 + 1.04i)13-s + (3.01 − 1.84i)15-s + (−0.353 − 1.31i)17-s + (1.55 + 2.70i)19-s + (−4.17 + 0.280i)21-s + (−4.44 − 1.18i)23-s + (4.19 − 2.71i)25-s + (−3.91 + 3.91i)27-s + 7.90i·29-s + (−7.63 − 4.40i)31-s + ⋯
L(s)  = 1  + (0.881 − 0.236i)3-s + (0.959 − 0.283i)5-s + (−0.981 − 0.193i)7-s + (−0.144 + 0.0835i)9-s + (−0.103 + 0.178i)11-s + (0.290 + 0.290i)13-s + (0.778 − 0.476i)15-s + (−0.0857 − 0.320i)17-s + (0.357 + 0.619i)19-s + (−0.910 + 0.0611i)21-s + (−0.925 − 0.248i)23-s + (0.839 − 0.542i)25-s + (−0.753 + 0.753i)27-s + 1.46i·29-s + (−1.37 − 0.791i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.962 + 0.271i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.962 + 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40812 - 0.194603i\)
\(L(\frac12)\) \(\approx\) \(1.40812 - 0.194603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.14 + 0.632i)T \)
7 \( 1 + (2.59 + 0.512i)T \)
good3 \( 1 + (-1.52 + 0.409i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.342 - 0.593i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.04 - 1.04i)T + 13iT^{2} \)
17 \( 1 + (0.353 + 1.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.55 - 2.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.44 + 1.18i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.90iT - 29T^{2} \)
31 \( 1 + (7.63 + 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 + 8.68i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (-3.73 + 3.73i)T - 43iT^{2} \)
47 \( 1 + (6.73 + 1.80i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.39 + 8.94i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.10 - 3.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.57 + 2.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.01 + 1.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (-12.4 + 3.32i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-12.1 + 7.01i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.99 - 3.99i)T + 83iT^{2} \)
89 \( 1 + (5.79 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.60 - 2.60i)T - 97iT^{2} \)
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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

−13.20040677626721778008140797686, −12.51545778971828008774514066209, −10.93570045417856098949802613766, −9.705565969691613500940191696688, −9.128568146315090855959965547990, −7.916446291379158000850818015442, −6.62546989372646228685798400934, −5.45073216016876998140421532493, −3.55731971226450848956354057207, −2.12833257748712173246151291087, 2.48879505025766592904648843031, 3.60485883141598160916923529719, 5.62465038603230807354140730562, 6.58811596514791954058802009168, 8.121372691503623877192723143319, 9.272876559683848350294344439210, 9.795642430215782067358467719207, 10.98562270550957986161232852328, 12.42813306675615793791630738826, 13.48927237587363693970156957421

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