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Properties

Label 2-143-143.32-c1-0-11
Degree $2$
Conductor $143$
Sign $0.746 + 0.665i$
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  + (2.13 − 0.571i)2-s + (0.785 − 1.36i)3-s + (2.48 − 1.43i)4-s + (−2.51 + 2.51i)5-s + (0.897 − 3.34i)6-s + (−1.32 − 0.354i)7-s + (1.35 − 1.35i)8-s + (0.264 + 0.458i)9-s + (−3.92 + 6.80i)10-s + (1.69 − 2.85i)11-s − 4.50i·12-s + (−3.36 + 1.30i)13-s − 3.02·14-s + (1.44 + 5.40i)15-s + (−0.756 + 1.30i)16-s + (−1.16 − 2.01i)17-s + ⋯
L(s)  = 1  + (1.50 − 0.403i)2-s + (0.453 − 0.785i)3-s + (1.24 − 0.716i)4-s + (−1.12 + 1.12i)5-s + (0.366 − 1.36i)6-s + (−0.499 − 0.133i)7-s + (0.478 − 0.478i)8-s + (0.0882 + 0.152i)9-s + (−1.24 + 2.15i)10-s + (0.510 − 0.860i)11-s − 1.30i·12-s + (−0.932 + 0.360i)13-s − 0.807·14-s + (0.373 + 1.39i)15-s + (−0.189 + 0.327i)16-s + (−0.282 − 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.746 + 0.665i$
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: \((2,\ 143,\ (\ :1/2),\ 0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00120 - 0.763122i\)
\(L(\frac12)\) \(\approx\) \(2.00120 - 0.763122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.36 - 1.30i)T \)
good2 \( 1 + (-2.13 + 0.571i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.785 + 1.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.51 - 2.51i)T - 5iT^{2} \)
7 \( 1 + (1.32 + 0.354i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (1.16 + 2.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 + 4.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.45 - 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.90 - 3.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.674 + 0.674i)T - 31iT^{2} \)
37 \( 1 + (5.07 - 1.36i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.91 - 0.514i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.79 + 3.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.14 + 8.14i)T + 47iT^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (1.54 - 5.75i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.41 + 0.816i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.108 - 0.403i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.68 + 0.720i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.5 - 11.5i)T - 73iT^{2} \)
79 \( 1 + 2.10iT - 79T^{2} \)
83 \( 1 + (-11.5 - 11.5i)T + 83iT^{2} \)
89 \( 1 + (-11.7 + 3.15i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.77 - 0.474i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.19068507984141231804936572988, −12.00023738516219527092327648473, −11.51477122709327512617312066943, −10.44137941594177844725381255390, −8.594207525947282979311217994601, −7.07580585277572953420994497887, −6.74711800742431359697181483179, −4.86929327178290178485794054350, −3.47010257041443239853888127383, −2.66297408745460257493262283957, 3.31192891160007144234884150187, 4.31082824319011891917507585169, 4.90729132732137447116442853142, 6.51986414646976036454830055258, 7.80609150494468170130339890373, 9.062773523160070814876760108743, 10.07381289154892994844630491652, 11.97936596613280858890543820881, 12.29150599337141070164284935813, 13.10881821072162672916485133022

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