Properties

Label 2-140-7.6-c14-0-35
Degree $2$
Conductor $140$
Sign $-0.335 - 0.942i$
Analytic cond. $174.060$
Root an. cond. $13.1932$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08e3i·3-s − 3.49e4i·5-s + (7.75e5 − 2.76e5i)7-s − 4.76e6·9-s − 1.57e7·11-s − 7.52e7i·13-s − 1.07e8·15-s − 5.31e8i·17-s + 6.79e8i·19-s + (−8.53e8 − 2.39e9i)21-s + 2.16e9·23-s − 1.22e9·25-s − 5.62e7i·27-s − 3.70e9·29-s − 3.57e9i·31-s + ⋯
L(s)  = 1  − 1.41i·3-s − 0.447i·5-s + (0.942 − 0.335i)7-s − 0.996·9-s − 0.807·11-s − 1.19i·13-s − 0.631·15-s − 1.29i·17-s + 0.760i·19-s + (−0.473 − 1.33i)21-s + 0.634·23-s − 0.199·25-s − 0.00537i·27-s − 0.214·29-s − 0.129i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.335 - 0.942i$
Analytic conductor: \(174.060\)
Root analytic conductor: \(13.1932\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :7),\ -0.335 - 0.942i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.336835244\)
\(L(\frac12)\) \(\approx\) \(1.336835244\)
\(L(8)\) 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 \( 1 + 3.49e4iT \)
7 \( 1 + (-7.75e5 + 2.76e5i)T \)
good3 \( 1 + 3.08e3iT - 4.78e6T^{2} \)
11 \( 1 + 1.57e7T + 3.79e14T^{2} \)
13 \( 1 + 7.52e7iT - 3.93e15T^{2} \)
17 \( 1 + 5.31e8iT - 1.68e17T^{2} \)
19 \( 1 - 6.79e8iT - 7.99e17T^{2} \)
23 \( 1 - 2.16e9T + 1.15e19T^{2} \)
29 \( 1 + 3.70e9T + 2.97e20T^{2} \)
31 \( 1 + 3.57e9iT - 7.56e20T^{2} \)
37 \( 1 - 5.03e9T + 9.01e21T^{2} \)
41 \( 1 - 2.33e11iT - 3.79e22T^{2} \)
43 \( 1 + 2.46e11T + 7.38e22T^{2} \)
47 \( 1 + 4.68e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.75e11T + 1.37e24T^{2} \)
59 \( 1 + 3.59e12iT - 6.19e24T^{2} \)
61 \( 1 + 3.47e12iT - 9.87e24T^{2} \)
67 \( 1 + 5.66e12T + 3.67e25T^{2} \)
71 \( 1 - 7.81e10T + 8.27e25T^{2} \)
73 \( 1 - 7.35e12iT - 1.22e26T^{2} \)
79 \( 1 + 2.41e13T + 3.68e26T^{2} \)
83 \( 1 - 7.63e12iT - 7.36e26T^{2} \)
89 \( 1 + 1.57e13iT - 1.95e27T^{2} \)
97 \( 1 - 6.35e13iT - 6.52e27T^{2} \)
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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

−9.896941468288032139158644098989, −8.336892780707373647791452344703, −7.83723258824294672566268777074, −6.99781136989533162855810877540, −5.63071912147599155040008116224, −4.80778035687504440612113374423, −3.07130814317196598289570948481, −1.94778993896917799321834286664, −1.01487181407838478739747565419, −0.26092083434820886548470582979, 1.63071330459399913818046695994, 2.80058596088372328875153144879, 4.04530446956561968880624171898, 4.77668486725455012121501375516, 5.77840295818668902137508128764, 7.20376496826871471407647072646, 8.501613142262367805869334123505, 9.263632546595104745786467932706, 10.46018914030753059899597115971, 10.93568523172066346021630452313

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