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Properties

Label 2-143-143.32-c1-0-0
Degree $2$
Conductor $143$
Sign $-0.657 + 0.753i$
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  + (−1.33 + 0.358i)2-s + (−1.38 + 2.39i)3-s + (−0.0732 + 0.0423i)4-s + (−0.860 + 0.860i)5-s + (0.990 − 3.69i)6-s + (−1.31 − 0.351i)7-s + (2.04 − 2.04i)8-s + (−2.32 − 4.02i)9-s + (0.842 − 1.45i)10-s + (3.31 + 0.189i)11-s − 0.233i·12-s + (−3.58 − 0.353i)13-s + 1.88·14-s + (−0.870 − 3.25i)15-s + (−1.91 + 3.31i)16-s + (−2.93 − 5.08i)17-s + ⋯
L(s)  = 1  + (−0.945 + 0.253i)2-s + (−0.798 + 1.38i)3-s + (−0.0366 + 0.0211i)4-s + (−0.384 + 0.384i)5-s + (0.404 − 1.50i)6-s + (−0.496 − 0.132i)7-s + (0.721 − 0.721i)8-s + (−0.773 − 1.34i)9-s + (0.266 − 0.461i)10-s + (0.998 + 0.0572i)11-s − 0.0675i·12-s + (−0.995 − 0.0980i)13-s + 0.502·14-s + (−0.224 − 0.839i)15-s + (−0.477 + 0.827i)16-s + (−0.711 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0680969 - 0.149845i\)
\(L(\frac12)\) \(\approx\) \(0.0680969 - 0.149845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.31 - 0.189i)T \)
13 \( 1 + (3.58 + 0.353i)T \)
good2 \( 1 + (1.33 - 0.358i)T + (1.73 - i)T^{2} \)
3 \( 1 + (1.38 - 2.39i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.860 - 0.860i)T - 5iT^{2} \)
7 \( 1 + (1.31 + 0.351i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (2.93 + 5.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.95 - 7.30i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.03 + 2.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.885 - 0.511i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.50 + 2.50i)T - 31iT^{2} \)
37 \( 1 + (4.77 - 1.27i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.42 - 0.650i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.853 - 1.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.54 - 3.54i)T + 47iT^{2} \)
53 \( 1 + 0.330T + 53T^{2} \)
59 \( 1 + (1.27 - 4.76i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.79 - 5.65i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.26 + 4.72i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.68 - 0.987i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.47 - 6.47i)T - 73iT^{2} \)
79 \( 1 - 0.0534iT - 79T^{2} \)
83 \( 1 + (-6.77 - 6.77i)T + 83iT^{2} \)
89 \( 1 + (12.4 - 3.33i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-17.6 - 4.73i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.99961963750891318802299093766, −12.37750220438228676339874622716, −11.46522376746820341953975408570, −10.29389984356501447469182153233, −9.802712414055185070279504984740, −8.916644199598212730177724763728, −7.47853655914429424948445700443, −6.30684443550443727100952679921, −4.66852098410925136039626713574, −3.71461499490293899565467507276, 0.24389688378953175509374486275, 1.91051338303683191029526218503, 4.64804807888958188111711018483, 6.21943448822492264093859054895, 7.14448302762058492459336846268, 8.311400636537556675728858144869, 9.207550646223053819570486173660, 10.50718614949621776063995465275, 11.57916862283679113597022071655, 12.28692427810430699595103379411

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