Properties

Label 2-143-1.1-c1-0-9
Degree $2$
Conductor $143$
Sign $1$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s + 3.30·3-s − 0.135·4-s − 3.38·5-s + 4.51·6-s − 1.56·7-s − 2.91·8-s + 7.94·9-s − 4.61·10-s − 11-s − 0.449·12-s + 13-s − 2.13·14-s − 11.1·15-s − 3.71·16-s − 2.73·17-s + 10.8·18-s + 4.69·19-s + 0.459·20-s − 5.17·21-s − 1.36·22-s + 4.67·23-s − 9.64·24-s + 6.43·25-s + 1.36·26-s + 16.3·27-s + 0.212·28-s + ⋯
L(s)  = 1  + 0.965·2-s + 1.91·3-s − 0.0678·4-s − 1.51·5-s + 1.84·6-s − 0.591·7-s − 1.03·8-s + 2.64·9-s − 1.46·10-s − 0.301·11-s − 0.129·12-s + 0.277·13-s − 0.571·14-s − 2.88·15-s − 0.927·16-s − 0.662·17-s + 2.55·18-s + 1.07·19-s + 0.102·20-s − 1.13·21-s − 0.291·22-s + 0.974·23-s − 1.96·24-s + 1.28·25-s + 0.267·26-s + 3.15·27-s + 0.0401·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.061556825\)
\(L(\frac12)\) \(\approx\) \(2.061556825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
13 \( 1 - T \)
good2 \( 1 - 1.36T + 2T^{2} \)
3 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 + 3.38T + 5T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
17 \( 1 + 2.73T + 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 - 4.67T + 23T^{2} \)
29 \( 1 + 2.34T + 29T^{2} \)
31 \( 1 + 1.16T + 31T^{2} \)
37 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 + 5.83T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 + 1.88T + 47T^{2} \)
53 \( 1 + 7.54T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 9.83T + 61T^{2} \)
67 \( 1 - 0.889T + 67T^{2} \)
71 \( 1 - 2.62T + 71T^{2} \)
73 \( 1 - 8.80T + 73T^{2} \)
79 \( 1 + 0.345T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 3.54T + 89T^{2} \)
97 \( 1 - 9.78T + 97T^{2} \)
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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

−13.19986949618191502842067181190, −12.65054970192919184270840965020, −11.39717908994826648457689192219, −9.694786525040283198570775864335, −8.805565386789896337685252414950, −7.936694032385145264926934670794, −6.90881314276337715121901318903, −4.69772697039292234229354620349, −3.64663246392807405313829053669, −3.02901126326435583138680693038, 3.02901126326435583138680693038, 3.64663246392807405313829053669, 4.69772697039292234229354620349, 6.90881314276337715121901318903, 7.936694032385145264926934670794, 8.805565386789896337685252414950, 9.694786525040283198570775864335, 11.39717908994826648457689192219, 12.65054970192919184270840965020, 13.19986949618191502842067181190

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