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Properties

Label 2-143-11.3-c1-0-11
Degree $2$
Conductor $143$
Sign $-0.150 + 0.988i$
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  + (1.61 − 1.17i)2-s + (−0.781 − 2.40i)3-s + (0.610 − 1.87i)4-s + (1.47 + 1.07i)5-s + (−4.07 − 2.96i)6-s + (−0.624 + 1.92i)7-s + (0.0147 + 0.0453i)8-s + (−2.74 + 1.99i)9-s + 3.64·10-s + (−2.99 − 1.41i)11-s − 4.99·12-s + (−0.809 + 0.587i)13-s + (1.24 + 3.83i)14-s + (1.42 − 4.39i)15-s + (3.27 + 2.37i)16-s + (1.34 + 0.975i)17-s + ⋯
L(s)  = 1  + (1.14 − 0.828i)2-s + (−0.451 − 1.38i)3-s + (0.305 − 0.939i)4-s + (0.660 + 0.480i)5-s + (−1.66 − 1.20i)6-s + (−0.236 + 0.726i)7-s + (0.00520 + 0.0160i)8-s + (−0.915 + 0.665i)9-s + 1.15·10-s + (−0.904 − 0.427i)11-s − 1.44·12-s + (−0.224 + 0.163i)13-s + (0.333 + 1.02i)14-s + (0.368 − 1.13i)15-s + (0.818 + 0.594i)16-s + (0.325 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.150 + 0.988i$
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: \((2,\ 143,\ (\ :1/2),\ -0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12153 - 1.30536i\)
\(L(\frac12)\) \(\approx\) \(1.12153 - 1.30536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.99 + 1.41i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-1.61 + 1.17i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.781 + 2.40i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.47 - 1.07i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.624 - 1.92i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-1.34 - 0.975i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.610 + 1.87i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.93T + 23T^{2} \)
29 \( 1 + (2.68 - 8.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.14 + 1.55i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.429 + 1.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.86 + 8.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + (1.90 + 5.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.98 - 7.25i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.23 - 3.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-12.5 - 9.14i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.46T + 67T^{2} \)
71 \( 1 + (2.88 + 2.09i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.140 + 0.431i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.90 + 2.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.13 + 4.45i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.42T + 89T^{2} \)
97 \( 1 + (-13.8 + 10.0i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.85923870562892711128554223652, −12.13063658012053619204097620501, −11.24012220983244057590470190143, −10.27011102182949079373486988654, −8.546162606640065723392220130109, −7.12222492494836737979713826833, −5.99764859764990921010779399491, −5.20699692199457465555116765609, −3.02479825304038861625980914620, −1.98142626856257195180650612984, 3.53000427333258338898116106298, 4.79323156293915179886524420206, 5.26037943019858314396334981716, 6.48219359035966900839264917444, 7.88369398183383774211262988229, 9.796069969653974892988603635370, 9.999618890235605082617960173654, 11.38492956363756019850284824789, 12.87929451691520890279888343325, 13.39673466444073672315559953304

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