Properties

Label 2-143-1.1-c3-0-5
Degree $2$
Conductor $143$
Sign $1$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.76·2-s − 4.70·3-s + 6.16·4-s + 20.6·5-s + 17.7·6-s − 28.6·7-s + 6.91·8-s − 4.84·9-s − 77.5·10-s − 11·11-s − 29.0·12-s − 13·13-s + 107.·14-s − 96.9·15-s − 75.3·16-s + 50.0·17-s + 18.2·18-s + 5.38·19-s + 126.·20-s + 134.·21-s + 41.3·22-s + 183.·23-s − 32.5·24-s + 299.·25-s + 48.9·26-s + 149.·27-s − 176.·28-s + ⋯
L(s)  = 1  − 1.33·2-s − 0.905·3-s + 0.770·4-s + 1.84·5-s + 1.20·6-s − 1.54·7-s + 0.305·8-s − 0.179·9-s − 2.45·10-s − 0.301·11-s − 0.697·12-s − 0.277·13-s + 2.05·14-s − 1.66·15-s − 1.17·16-s + 0.714·17-s + 0.238·18-s + 0.0649·19-s + 1.41·20-s + 1.40·21-s + 0.401·22-s + 1.66·23-s − 0.276·24-s + 2.39·25-s + 0.369·26-s + 1.06·27-s − 1.19·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $1$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6106952373\)
\(L(\frac12)\) \(\approx\) \(0.6106952373\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11T \)
13 \( 1 + 13T \)
good2 \( 1 + 3.76T + 8T^{2} \)
3 \( 1 + 4.70T + 27T^{2} \)
5 \( 1 - 20.6T + 125T^{2} \)
7 \( 1 + 28.6T + 343T^{2} \)
17 \( 1 - 50.0T + 4.91e3T^{2} \)
19 \( 1 - 5.38T + 6.85e3T^{2} \)
23 \( 1 - 183.T + 1.21e4T^{2} \)
29 \( 1 + 200.T + 2.43e4T^{2} \)
31 \( 1 + 19.4T + 2.97e4T^{2} \)
37 \( 1 - 258.T + 5.06e4T^{2} \)
41 \( 1 - 359.T + 6.89e4T^{2} \)
43 \( 1 - 261.T + 7.95e4T^{2} \)
47 \( 1 + 450.T + 1.03e5T^{2} \)
53 \( 1 - 725.T + 1.48e5T^{2} \)
59 \( 1 - 381.T + 2.05e5T^{2} \)
61 \( 1 - 68.9T + 2.26e5T^{2} \)
67 \( 1 - 133.T + 3.00e5T^{2} \)
71 \( 1 + 142.T + 3.57e5T^{2} \)
73 \( 1 - 394.T + 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 519.T + 5.71e5T^{2} \)
89 \( 1 - 248.T + 7.04e5T^{2} \)
97 \( 1 - 848.T + 9.12e5T^{2} \)
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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

−12.79872557901009462149002972624, −11.14718439740545620448512178568, −10.26408040770734570476139762547, −9.605145464597633201921554802636, −8.987485766549773729195495962588, −7.16783499178004325396712809682, −6.18643580730669169585960002866, −5.34044784485604815028863313873, −2.63249420337107675747850241871, −0.817296351662831924339291380066, 0.817296351662831924339291380066, 2.63249420337107675747850241871, 5.34044784485604815028863313873, 6.18643580730669169585960002866, 7.16783499178004325396712809682, 8.987485766549773729195495962588, 9.605145464597633201921554802636, 10.26408040770734570476139762547, 11.14718439740545620448512178568, 12.79872557901009462149002972624

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