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Properties

Label 2-143-11.4-c1-0-5
Degree $2$
Conductor $143$
Sign $-0.970 + 0.242i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 1.31i)2-s + (−1 + 3.07i)3-s + (0.927 + 2.85i)4-s + (−1 + 0.726i)5-s + (5.85 − 4.25i)6-s + (−0.881 − 2.71i)7-s + (0.690 − 2.12i)8-s + (−6.04 − 4.39i)9-s + 2.76·10-s + (−1.69 − 2.85i)11-s − 9.70·12-s + (0.809 + 0.587i)13-s + (−1.97 + 6.06i)14-s + (−1.23 − 3.80i)15-s + (0.809 − 0.587i)16-s + (−1.92 + 1.40i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.929i)2-s + (−0.577 + 1.77i)3-s + (0.463 + 1.42i)4-s + (−0.447 + 0.324i)5-s + (2.38 − 1.73i)6-s + (−0.333 − 1.02i)7-s + (0.244 − 0.751i)8-s + (−2.01 − 1.46i)9-s + 0.874·10-s + (−0.509 − 0.860i)11-s − 2.80·12-s + (0.224 + 0.163i)13-s + (−0.527 + 1.62i)14-s + (−0.319 − 0.982i)15-s + (0.202 − 0.146i)16-s + (−0.467 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.69 + 2.85i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (1.80 + 1.31i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (1 - 3.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1 - 0.726i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.881 + 2.71i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (1.92 - 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.190 + 0.587i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (1.04 + 3.21i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.73 + 1.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.23 - 3.80i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2.88 - 8.86i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.97 - 6.51i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.28 + 7.02i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.73 - 4.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (6.92 - 5.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.38 + 7.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.61 - 1.17i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.5 + 1.81i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (3 + 2.17i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.85027021522541315783653540136, −11.13461514532057168231218931270, −10.53855271898202589411291805790, −9.903026383810749241808287150322, −8.945685282148402767329204512107, −7.81316691852144517603969583943, −5.94856886312726049025137337353, −4.18530991111405147645135492659, −3.25163896954720045038248574394, 0, 2.01681471401067289567481233075, 5.50294201377845037711479248224, 6.39585392802704992046250028916, 7.35507579265436809560523909046, 8.107300246094243817245360800607, 8.936732034713977708398467715102, 10.36858497598610371150644166400, 11.84970807290345079521322293259, 12.39078365182653492365100909662

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