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Properties

Label 2-143-11.3-c1-0-4
Degree $2$
Conductor $143$
Sign $0.672 - 0.740i$
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  + (−2.04 + 1.48i)2-s + (0.0248 + 0.0764i)3-s + (1.35 − 4.18i)4-s + (1.00 + 0.733i)5-s + (−0.164 − 0.119i)6-s + (1.01 − 3.11i)7-s + (1.87 + 5.77i)8-s + (2.42 − 1.75i)9-s − 3.15·10-s + (1.71 + 2.84i)11-s + 0.353·12-s + (−0.809 + 0.587i)13-s + (2.56 + 7.88i)14-s + (−0.0309 + 0.0953i)15-s + (−5.31 − 3.85i)16-s + (−2.46 − 1.78i)17-s + ⋯
L(s)  = 1  + (−1.44 + 1.05i)2-s + (0.0143 + 0.0441i)3-s + (0.679 − 2.09i)4-s + (0.451 + 0.328i)5-s + (−0.0671 − 0.0487i)6-s + (0.382 − 1.17i)7-s + (0.663 + 2.04i)8-s + (0.807 − 0.586i)9-s − 0.998·10-s + (0.516 + 0.856i)11-s + 0.102·12-s + (−0.224 + 0.163i)13-s + (0.684 + 2.10i)14-s + (−0.00800 + 0.0246i)15-s + (−1.32 − 0.964i)16-s + (−0.597 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.672 - 0.740i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.672 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607464 + 0.268758i\)
\(L(\frac12)\) \(\approx\) \(0.607464 + 0.268758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.71 - 2.84i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (2.04 - 1.48i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.0248 - 0.0764i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.00 - 0.733i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.01 + 3.11i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (2.46 + 1.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.33 - 7.18i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.15T + 23T^{2} \)
29 \( 1 + (-0.753 + 2.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.64 + 2.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.697 + 2.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.955 + 2.94i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + (-1.12 - 3.47i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.41 - 4.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.48 - 13.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.12 - 1.54i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + (10.7 + 7.83i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.66 + 5.11i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.50 + 3.27i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.46 - 1.79i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (-6.10 + 4.43i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.65498783469823411588558012159, −12.03717780466764640633697542652, −10.55240841835756617353687055804, −9.980728997329126260460762851455, −9.201267642387454226806670911479, −7.74534558646557833739843914875, −7.09029736886937790743664180715, −6.19883113171327304489595147223, −4.38849487790998146255242248665, −1.41638104378077829025434411737, 1.55400942304101952694734418285, 2.90522633763700025688274648489, 5.05177680510066366515253172706, 6.88461512376218176448589965853, 8.293785507484743208268049430499, 8.961461428951130955709822081277, 9.755573085408621004057236050369, 10.98753303772856767299254606346, 11.58042453971701814735803527235, 12.69984498628644089846569664690

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