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Properties

Label 2-140-35.9-c1-0-3
Degree $2$
Conductor $140$
Sign $0.669 + 0.742i$
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  + (1.5 − 0.866i)3-s + (−0.5 − 2.17i)5-s + (−1.13 − 2.38i)7-s + (2.63 + 4.56i)11-s + 2.62i·13-s + (−2.63 − 2.83i)15-s + (−0.362 + 0.209i)17-s + (1.63 − 2.83i)19-s + (−3.77 − 2.59i)21-s + (6.77 + 3.91i)23-s + (−4.50 + 2.17i)25-s + 5.19i·27-s − 4.27·29-s + (−1.63 − 2.83i)31-s + (7.91 + 4.56i)33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.223 − 0.974i)5-s + (−0.429 − 0.902i)7-s + (0.795 + 1.37i)11-s + 0.728i·13-s + (−0.680 − 0.732i)15-s + (−0.0879 + 0.0507i)17-s + (0.375 − 0.650i)19-s + (−0.823 − 0.566i)21-s + (1.41 + 0.815i)23-s + (−0.900 + 0.435i)25-s + 0.999i·27-s − 0.793·29-s + (−0.294 − 0.509i)31-s + (1.37 + 0.795i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20551 - 0.536547i\)
\(L(\frac12)\) \(\approx\) \(1.20551 - 0.536547i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.17i)T \)
7 \( 1 + (1.13 + 2.38i)T \)
good3 \( 1 + (-1.5 + 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.62iT - 13T^{2} \)
17 \( 1 + (0.362 - 0.209i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.63 + 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 + 2.15iT - 43T^{2} \)
47 \( 1 + (-5.63 - 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.63 + 2.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.04 - 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 + (-5.63 + 3.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.63 + 6.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.40iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−13.13874524576967597113312175109, −12.29280575824357261179668269929, −11.09122549690287877617055865588, −9.486777281139055447548499103507, −9.016022891953113342712542735998, −7.56724516418755785646199133188, −6.97272893606860122951055659982, −4.94560266360483320586645260055, −3.71715217461558743430330436490, −1.70187967642289420603536752132, 2.92974880914809004783737323761, 3.55780134359169229725739791947, 5.67768649321110556795598673553, 6.78512934521630292500291983834, 8.352237345153777076379897922362, 9.006782490216052697291878239909, 10.14226387607678464297435538674, 11.21232054841931837271673296354, 12.19584353081240943480137896528, 13.55106033424159937871462411124

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