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Properties

Label 2-143-11.5-c1-0-2
Degree $2$
Conductor $143$
Sign $-0.687 - 0.725i$
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  + (0.760 + 2.34i)2-s + (0.674 + 0.489i)3-s + (−3.28 + 2.38i)4-s + (−0.304 + 0.937i)5-s + (−0.633 + 1.95i)6-s + (3.14 − 2.28i)7-s + (−4.09 − 2.97i)8-s + (−0.712 − 2.19i)9-s − 2.42·10-s + (−3.26 + 0.581i)11-s − 3.37·12-s + (−0.309 − 0.951i)13-s + (7.73 + 5.62i)14-s + (−0.664 + 0.482i)15-s + (1.34 − 4.12i)16-s + (0.674 − 2.07i)17-s + ⋯
L(s)  = 1  + (0.537 + 1.65i)2-s + (0.389 + 0.282i)3-s + (−1.64 + 1.19i)4-s + (−0.136 + 0.419i)5-s + (−0.258 + 0.796i)6-s + (1.18 − 0.863i)7-s + (−1.44 − 1.05i)8-s + (−0.237 − 0.730i)9-s − 0.767·10-s + (−0.984 + 0.175i)11-s − 0.975·12-s + (−0.0857 − 0.263i)13-s + (2.06 + 1.50i)14-s + (−0.171 + 0.124i)15-s + (0.335 − 1.03i)16-s + (0.163 − 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.687 - 0.725i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.687 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589450 + 1.37099i\)
\(L(\frac12)\) \(\approx\) \(0.589450 + 1.37099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.26 - 0.581i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.760 - 2.34i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.674 - 0.489i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.304 - 0.937i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.14 + 2.28i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-0.674 + 2.07i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.27 - 3.10i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 + (5.51 - 4.00i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.01 + 3.13i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.73 + 5.61i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.77 + 2.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.11T + 43T^{2} \)
47 \( 1 + (1.55 + 1.13i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.42 - 4.39i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.0353 - 0.0256i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.28 - 10.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.77T + 67T^{2} \)
71 \( 1 + (-3.25 + 10.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.9 - 9.42i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.87 + 8.83i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.79 + 11.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.97T + 89T^{2} \)
97 \( 1 + (1.73 + 5.32i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.95827906440153132424529977620, −12.99992749437718342679436884166, −11.57153038300223144625019517380, −10.28088304064137685138560649931, −8.920307051842541360289965312712, −7.69757294520922924780909424040, −7.38803836840366938741160169457, −5.80771109515914217634427967231, −4.75943929604903176277474146286, −3.51930426286971661882104588516, 1.79748126451848251019508160565, 2.88125841935139321114024472878, 4.70016776528929676547597389744, 5.38558785214223360676746056451, 7.890623958283720748309931321010, 8.680628015986522048221368865692, 9.938424493068959053759224754377, 11.12564702333315843969868188017, 11.63314185609625077739406849648, 12.69275530792367196542784740766

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