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Properties

Label 2-140-7.6-c14-0-14
Degree $2$
Conductor $140$
Sign $-0.863 - 0.504i$
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  − 303. i·3-s + 3.49e4i·5-s + (−4.15e5 + 7.11e5i)7-s + 4.69e6·9-s + 1.67e7·11-s + 1.13e8i·13-s + 1.06e7·15-s + 7.03e8i·17-s + 5.27e8i·19-s + (2.15e8 + 1.26e8i)21-s − 3.75e9·23-s − 1.22e9·25-s − 2.87e9i·27-s + 7.44e9·29-s + 4.18e10i·31-s + ⋯
L(s)  = 1  − 0.138i·3-s + 0.447i·5-s + (−0.504 + 0.863i)7-s + 0.980·9-s + 0.859·11-s + 1.81i·13-s + 0.0620·15-s + 1.71i·17-s + 0.589i·19-s + (0.119 + 0.0700i)21-s − 1.10·23-s − 0.199·25-s − 0.274i·27-s + 0.431·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.863 - 0.504i$
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: \((2,\ 140,\ (\ :7),\ -0.863 - 0.504i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(2.268730213\)
\(L(\frac12)\) \(\approx\) \(2.268730213\)
\(L(8)\) 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 \( 1 + (4.15e5 - 7.11e5i)T \)
good3 \( 1 + 303. iT - 4.78e6T^{2} \)
11 \( 1 - 1.67e7T + 3.79e14T^{2} \)
13 \( 1 - 1.13e8iT - 3.93e15T^{2} \)
17 \( 1 - 7.03e8iT - 1.68e17T^{2} \)
19 \( 1 - 5.27e8iT - 7.99e17T^{2} \)
23 \( 1 + 3.75e9T + 1.15e19T^{2} \)
29 \( 1 - 7.44e9T + 2.97e20T^{2} \)
31 \( 1 - 4.18e10iT - 7.56e20T^{2} \)
37 \( 1 - 1.57e11T + 9.01e21T^{2} \)
41 \( 1 + 1.56e10iT - 3.79e22T^{2} \)
43 \( 1 - 3.48e11T + 7.38e22T^{2} \)
47 \( 1 - 5.96e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.35e12T + 1.37e24T^{2} \)
59 \( 1 + 2.06e11iT - 6.19e24T^{2} \)
61 \( 1 + 5.42e12iT - 9.87e24T^{2} \)
67 \( 1 + 3.85e12T + 3.67e25T^{2} \)
71 \( 1 - 1.27e13T + 8.27e25T^{2} \)
73 \( 1 + 5.67e12iT - 1.22e26T^{2} \)
79 \( 1 - 2.24e13T + 3.68e26T^{2} \)
83 \( 1 - 8.87e12iT - 7.36e26T^{2} \)
89 \( 1 + 5.93e13iT - 1.95e27T^{2} \)
97 \( 1 - 2.97e13iT - 6.52e27T^{2} \)
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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

−10.99853895849208295651979141626, −9.837067402725183896668633507978, −9.106988792925720179614152049009, −7.909684283986000999127117895021, −6.49968565946847836630586882403, −6.25034001562252605465232528650, −4.42092790963092128443584398769, −3.63205609170766884661537029046, −2.09615659084227893315169531964, −1.41947490237912237601728851879, 0.50974187987669089069062025602, 0.938799567666015940450308011539, 2.61036578504982359379994871346, 3.83529936486656104617764627018, 4.67531122912577060145586364006, 5.95995411930423049081554655833, 7.16381389564760076691434126102, 7.921730166943333993454334465104, 9.450886274514263424371791543599, 9.931482708596419196254715480448

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