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Properties

Label 2-143-13.9-c1-0-9
Degree $2$
Conductor $143$
Sign $0.0128 + 0.999i$
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  + (1.14 − 1.97i)2-s + (−0.610 + 1.05i)3-s + (−1.61 − 2.79i)4-s + 2.50·5-s + (1.39 + 2.41i)6-s + (−2.25 − 3.90i)7-s − 2.79·8-s + (0.753 + 1.30i)9-s + (2.86 − 4.96i)10-s + (−0.5 + 0.866i)11-s + 3.93·12-s + (−2.5 + 2.59i)13-s − 10.2·14-s + (−1.53 + 2.65i)15-s + (0.0316 − 0.0547i)16-s + (1.61 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.807 − 1.39i)2-s + (−0.352 + 0.610i)3-s + (−0.805 − 1.39i)4-s + 1.12·5-s + (0.569 + 0.987i)6-s + (−0.851 − 1.47i)7-s − 0.987·8-s + (0.251 + 0.435i)9-s + (0.905 − 1.56i)10-s + (−0.150 + 0.261i)11-s + 1.13·12-s + (−0.693 + 0.720i)13-s − 2.75·14-s + (−0.395 + 0.684i)15-s + (0.00790 − 0.0136i)16-s + (0.390 + 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09989 - 1.08588i\)
\(L(\frac12)\) \(\approx\) \(1.09989 - 1.08588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (-1.14 + 1.97i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.610 - 1.05i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.50T + 5T^{2} \)
7 \( 1 + (2.25 + 3.90i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-1.61 - 2.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 2.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.89 - 5.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.309 + 0.536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 + (2.53 - 4.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.18 + 7.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.54 + 9.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + 4.06T + 53T^{2} \)
59 \( 1 + (5.53 + 9.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0722 - 0.125i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.36 - 7.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.857 - 1.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 1.87T + 79T^{2} \)
83 \( 1 + 5.38T + 83T^{2} \)
89 \( 1 + (-4.25 + 7.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0562 + 0.0974i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.92078433401637180533982538053, −11.88111653171697433850913843957, −10.62565807290168967071578380746, −10.04276939467740251449158686674, −9.716173520272395531272323110142, −7.32629102233358142052643440996, −5.79953328225383607209502418739, −4.59405422214158316108982665357, −3.61677235568673405868551834998, −1.84600934585149180624243899777, 2.78940096442065197352792970533, 5.03267913168695364035468188565, 6.03268270470242332116633811549, 6.36935453401415002536656161307, 7.68439637528586180533431846165, 9.049948998020525886885825402569, 9.989952929796305960170253565883, 12.01171249746395710739633641265, 12.69369532076517906283900919291, 13.36756094694110395029723012283

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