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Properties

Label 2-143-11.5-c1-0-1
Degree $2$
Conductor $143$
Sign $0.211 - 0.977i$
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  + (0.350 + 1.07i)2-s + (−0.0789 − 0.0573i)3-s + (0.578 − 0.420i)4-s + (−1.10 + 3.39i)5-s + (0.0341 − 0.105i)6-s + (−0.120 + 0.0876i)7-s + (2.48 + 1.80i)8-s + (−0.924 − 2.84i)9-s − 4.04·10-s + (2.32 + 2.36i)11-s − 0.0698·12-s + (0.309 + 0.951i)13-s + (−0.136 − 0.0993i)14-s + (0.281 − 0.204i)15-s + (−0.635 + 1.95i)16-s + (1.81 − 5.57i)17-s + ⋯
L(s)  = 1  + (0.247 + 0.762i)2-s + (−0.0456 − 0.0331i)3-s + (0.289 − 0.210i)4-s + (−0.493 + 1.51i)5-s + (0.0139 − 0.0429i)6-s + (−0.0456 + 0.0331i)7-s + (0.880 + 0.639i)8-s + (−0.308 − 0.948i)9-s − 1.27·10-s + (0.699 + 0.714i)11-s − 0.0201·12-s + (0.0857 + 0.263i)13-s + (−0.0365 − 0.0265i)14-s + (0.0728 − 0.0528i)15-s + (−0.158 + 0.489i)16-s + (0.439 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.211 - 0.977i$
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: \((2,\ 143,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00536 + 0.810790i\)
\(L(\frac12)\) \(\approx\) \(1.00536 + 0.810790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.32 - 2.36i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.350 - 1.07i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.0789 + 0.0573i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.10 - 3.39i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.120 - 0.0876i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-1.81 + 5.57i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.72 + 4.15i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 + (-3.08 + 2.24i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.0102 + 0.0314i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.02 + 4.37i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.63 + 4.82i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + (4.18 + 3.04i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.19 - 12.9i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.0 - 7.32i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.956 + 2.94i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.22T + 67T^{2} \)
71 \( 1 + (2.68 - 8.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.47 + 1.79i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.44 + 10.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.82 - 5.62i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (-1.64 - 5.06i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.81603555265039618834037612704, −12.08960324342182771724099691204, −11.35721365375901532670359061761, −10.43545897528890895621297721342, −9.171039392480331465360101321035, −7.51586019151378281998422201513, −6.81859758136474937174791716258, −6.12595666149943589877401196128, −4.32598668464400857257194465639, −2.68912463055124894882890533316, 1.60017552837683230579749367350, 3.63005522035775970606863070946, 4.68273498435786282852406942035, 6.17091123054423708390865509740, 8.077951234197116141788672522915, 8.433269410392772638841206394128, 10.09759718902520295942870266516, 11.07833893836862447799394611458, 11.99779577090973272030930553311, 12.74764785955705472188444965665

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