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Properties

Label 2-143-11.3-c1-0-1
Degree $2$
Conductor $143$
Sign $0.327 - 0.944i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 + 0.243i)2-s + (−0.228 − 0.704i)3-s + (−0.564 + 1.73i)4-s + (0.645 + 0.468i)5-s + (0.248 + 0.180i)6-s + (−1.50 + 4.63i)7-s + (−0.490 − 1.51i)8-s + (1.98 − 1.44i)9-s − 0.331·10-s + (2.69 + 1.93i)11-s + 1.35·12-s + (−0.809 + 0.587i)13-s + (−0.625 − 1.92i)14-s + (0.182 − 0.562i)15-s + (−2.42 − 1.76i)16-s + (2.44 + 1.77i)17-s + ⋯
L(s)  = 1  + (−0.237 + 0.172i)2-s + (−0.132 − 0.406i)3-s + (−0.282 + 0.869i)4-s + (0.288 + 0.209i)5-s + (0.101 + 0.0737i)6-s + (−0.569 + 1.75i)7-s + (−0.173 − 0.534i)8-s + (0.660 − 0.480i)9-s − 0.104·10-s + (0.811 + 0.584i)11-s + 0.390·12-s + (−0.224 + 0.163i)13-s + (−0.167 − 0.514i)14-s + (0.0471 − 0.145i)15-s + (−0.605 − 0.440i)16-s + (0.592 + 0.430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.737614 + 0.524799i\)
\(L(\frac12)\) \(\approx\) \(0.737614 + 0.524799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.69 - 1.93i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.335 - 0.243i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.228 + 0.704i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.645 - 0.468i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.50 - 4.63i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-2.44 - 1.77i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.336 + 1.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.58T + 23T^{2} \)
29 \( 1 + (-0.373 + 1.14i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.99 + 3.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.15 + 3.55i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.50 + 4.63i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.58T + 43T^{2} \)
47 \( 1 + (-3.88 - 11.9i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.59 + 1.88i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.45 + 7.55i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.95 - 2.14i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.723T + 67T^{2} \)
71 \( 1 + (-1.12 - 0.816i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.84 - 8.75i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.7 - 8.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.2 - 7.45i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (4.99 - 3.62i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.94104866055545600224834272453, −12.29176035764527726311881906785, −11.83661491480682667084426015842, −9.799746916150030656466299852782, −9.237664544883088502199163395651, −8.102406818396287035827237676034, −6.82176173677338121824869344873, −5.96283331239494647109409097620, −4.08778699863523617947846427147, −2.44857827569322662068259503324, 1.15286201476894216916743183722, 3.80683658994193405026702725220, 4.96616350960555895407576065516, 6.35416738337203225468508795249, 7.54942689644321311391230279361, 9.128972511488763887853883667249, 10.18598478576505398747511249543, 10.36446757388739637637764735777, 11.65788359989537959897789200072, 13.29273092227396176229514601327

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