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Properties

Label 1-140-140.139-r0-0-0
Degree $1$
Conductor $140$
Sign $1$
Analytic cond. $0.650157$
Root an. cond. $0.650157$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s + 13-s + 17-s + 19-s + 23-s − 27-s + 29-s + 31-s + 33-s − 37-s − 39-s − 41-s + 43-s − 47-s − 51-s − 53-s − 57-s + 59-s − 61-s + 67-s − 69-s − 71-s + 73-s − 79-s + 81-s + ⋯
L(s)  = 1  − 3-s + 9-s − 11-s + 13-s + 17-s + 19-s + 23-s − 27-s + 29-s + 31-s + 33-s − 37-s − 39-s − 41-s + 43-s − 47-s − 51-s − 53-s − 57-s + 59-s − 61-s + 67-s − 69-s − 71-s + 73-s − 79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.650157\)
Root analytic conductor: \(0.650157\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{140} (139, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8279324314\)
\(L(\frac12)\) \(\approx\) \(0.8279324314\)
\(L(1)\) \(\approx\) \(0.8376792812\)
\(L(1)\) \(\approx\) \(0.8376792812\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.51413352346215424348953492723, −27.550413022031059179180179505480, −26.55700025407198435025394175357, −25.41430985356381423290187225014, −24.23362595960603611059204349195, −23.28266918225806378048740191633, −22.7019821752713791320521196144, −21.339602574332439356808704648188, −20.7435205737084878593512065036, −19.04900143075812866603920540428, −18.271449646817031966210138439886, −17.3217709353552537590723572666, −16.16078933964883395803517058248, −15.52222519051050917482954079172, −13.88351659347128372243297221218, −12.83038089878145821146427004523, −11.78055499774415879530259314605, −10.74966896648909606067154368601, −9.841445891456038745385021808676, −8.225708064914239414708430870124, −6.99271032358227376619997234292, −5.758530531850165769185477026681, −4.83877065214344767440406379388, −3.20988432931826998327617915013, −1.1743650171447352628831368674, 1.1743650171447352628831368674, 3.20988432931826998327617915013, 4.83877065214344767440406379388, 5.758530531850165769185477026681, 6.99271032358227376619997234292, 8.225708064914239414708430870124, 9.841445891456038745385021808676, 10.74966896648909606067154368601, 11.78055499774415879530259314605, 12.83038089878145821146427004523, 13.88351659347128372243297221218, 15.52222519051050917482954079172, 16.16078933964883395803517058248, 17.3217709353552537590723572666, 18.271449646817031966210138439886, 19.04900143075812866603920540428, 20.7435205737084878593512065036, 21.339602574332439356808704648188, 22.7019821752713791320521196144, 23.28266918225806378048740191633, 24.23362595960603611059204349195, 25.41430985356381423290187225014, 26.55700025407198435025394175357, 27.550413022031059179180179505480, 28.51413352346215424348953492723

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