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Properties

Label 2-143-143.98-c1-0-5
Degree $2$
Conductor $143$
Sign $0.351 - 0.936i$
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  + (0.428 + 1.60i)2-s + (0.382 − 0.663i)3-s + (−0.647 + 0.373i)4-s + (−0.104 − 0.104i)5-s + (1.22 + 0.328i)6-s + (−0.542 + 2.02i)7-s + (1.46 + 1.46i)8-s + (1.20 + 2.09i)9-s + (0.122 − 0.211i)10-s + (−1.26 − 3.06i)11-s + 0.571i·12-s + (2.52 − 2.57i)13-s − 3.47·14-s + (−0.109 + 0.0292i)15-s + (−2.46 + 4.27i)16-s + (−1.88 − 3.25i)17-s + ⋯
L(s)  = 1  + (0.303 + 1.13i)2-s + (0.221 − 0.382i)3-s + (−0.323 + 0.186i)4-s + (−0.0466 − 0.0466i)5-s + (0.500 + 0.134i)6-s + (−0.204 + 0.764i)7-s + (0.519 + 0.519i)8-s + (0.402 + 0.696i)9-s + (0.0386 − 0.0669i)10-s + (−0.381 − 0.924i)11-s + 0.165i·12-s + (0.701 − 0.713i)13-s − 0.928·14-s + (−0.0281 + 0.00754i)15-s + (−0.617 + 1.06i)16-s + (−0.456 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.351 - 0.936i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16261 + 0.805138i\)
\(L(\frac12)\) \(\approx\) \(1.16261 + 0.805138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.26 + 3.06i)T \)
13 \( 1 + (-2.52 + 2.57i)T \)
good2 \( 1 + (-0.428 - 1.60i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-0.382 + 0.663i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.104 + 0.104i)T + 5iT^{2} \)
7 \( 1 + (0.542 - 2.02i)T + (-6.06 - 3.5i)T^{2} \)
17 \( 1 + (1.88 + 3.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.23 + 1.40i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.14 + 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.93 + 2.84i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.15 - 7.15i)T + 31iT^{2} \)
37 \( 1 + (-0.877 - 3.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.88 - 7.02i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.18 + 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.00268 + 0.00268i)T - 47iT^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + (4.54 + 1.21i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-11.8 + 6.83i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.04 - 2.15i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.26 - 4.72i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.92 - 6.92i)T + 73iT^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + (-1.45 + 1.45i)T - 83iT^{2} \)
89 \( 1 + (0.243 + 0.908i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.57 + 5.87i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.53913532774901490363347456082, −12.69287849048272855945953953483, −11.28538974445189695714448744558, −10.30627010149057400124213514112, −8.471716551232125383937859775190, −8.132965945625475795098481934829, −6.69198930933515705495178824235, −5.87991183054573689618793877560, −4.65431885112900540031006661845, −2.48634265173730682870486208146, 1.87852870525591725360190025464, 3.76363470896644618058205478870, 4.25736481761383227699619648503, 6.43123956724021401834713355838, 7.58145941843346961790751774845, 9.207648272912551486804730860434, 10.12909236069638978739430769436, 10.83115032928989611199015544828, 11.91271011481002941478522222801, 12.87414299088573397815574549784

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