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Properties

Label 2-143-13.3-c1-0-8
Degree $2$
Conductor $143$
Sign $-0.293 + 0.956i$
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  + (−0.5 − 0.866i)2-s + (0.169 + 0.294i)3-s + (0.500 − 0.866i)4-s − 2.88·5-s + (0.169 − 0.294i)6-s + (1.77 − 3.06i)7-s − 3·8-s + (1.44 − 2.49i)9-s + (1.44 + 2.49i)10-s + (−0.5 − 0.866i)11-s + 0.339·12-s + (1.66 + 3.19i)13-s − 3.54·14-s + (−0.490 − 0.849i)15-s + (0.500 + 0.866i)16-s + (2.61 − 4.52i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0981 + 0.169i)3-s + (0.250 − 0.433i)4-s − 1.28·5-s + (0.0693 − 0.120i)6-s + (0.669 − 1.16i)7-s − 1.06·8-s + (0.480 − 0.832i)9-s + (0.456 + 0.789i)10-s + (−0.150 − 0.261i)11-s + 0.0981·12-s + (0.463 + 0.886i)13-s − 0.947·14-s + (−0.126 − 0.219i)15-s + (0.125 + 0.216i)16-s + (0.633 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.293 + 0.956i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.293 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513055 - 0.693881i\)
\(L(\frac12)\) \(\approx\) \(0.513055 - 0.693881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-1.66 - 3.19i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.169 - 0.294i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.88T + 5T^{2} \)
7 \( 1 + (-1.77 + 3.06i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-2.61 + 4.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.11 - 5.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.339 + 0.588i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.21 - 7.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.40T + 31T^{2} \)
37 \( 1 + (1.76 + 3.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.87 - 4.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.660 + 1.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.10T + 47T^{2} \)
53 \( 1 + 6.08T + 53T^{2} \)
59 \( 1 + (-7.05 + 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.00 - 3.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.05 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 - 0.904T + 79T^{2} \)
83 \( 1 - 4.90T + 83T^{2} \)
89 \( 1 + (-2.82 - 4.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.39 - 2.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.33278938642137365649833878983, −11.68390435241481062027274935686, −10.81240666346102103334470044089, −9.994727333803601536656683056767, −8.730484495730862413889235310425, −7.57562392098109479669587193035, −6.48910314984411861532895926726, −4.51667750288562432253196531162, −3.46295341241537849987082325893, −1.07096009591058075893692776287, 2.68195133408264526185760946624, 4.40597357128896291037508567201, 5.95184392495624034168651774020, 7.39295259010814437373898649618, 8.177888031923234930568388056919, 8.590147049804193543294919976648, 10.46358404282945214892883711119, 11.62167552872094285268642081951, 12.24505143956365416743257102912, 13.23226553800205213676080309910

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