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Properties

Label 2-143-143.25-c1-0-8
Degree $2$
Conductor $143$
Sign $-0.337 + 0.941i$
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  + (−0.510 − 0.703i)2-s + (−0.187 − 0.577i)3-s + (0.384 − 1.18i)4-s + (0.238 − 0.327i)5-s + (−0.310 + 0.426i)6-s + (−0.926 − 0.301i)7-s + (−2.68 + 0.871i)8-s + (2.12 − 1.54i)9-s − 0.352·10-s + (−0.972 − 3.17i)11-s − 0.755·12-s + (−1.07 + 3.44i)13-s + (0.261 + 0.805i)14-s + (−0.233 − 0.0760i)15-s + (−0.0312 − 0.0226i)16-s + (−2.49 − 1.81i)17-s + ⋯
L(s)  = 1  + (−0.361 − 0.497i)2-s + (−0.108 − 0.333i)3-s + (0.192 − 0.591i)4-s + (0.106 − 0.146i)5-s + (−0.126 + 0.174i)6-s + (−0.350 − 0.113i)7-s + (−0.948 + 0.308i)8-s + (0.709 − 0.515i)9-s − 0.111·10-s + (−0.293 − 0.956i)11-s − 0.218·12-s + (−0.299 + 0.954i)13-s + (0.0699 + 0.215i)14-s + (−0.0604 − 0.0196i)15-s + (−0.00780 − 0.00566i)16-s + (−0.604 − 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518331 - 0.736448i\)
\(L(\frac12)\) \(\approx\) \(0.518331 - 0.736448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.972 + 3.17i)T \)
13 \( 1 + (1.07 - 3.44i)T \)
good2 \( 1 + (0.510 + 0.703i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.187 + 0.577i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.238 + 0.327i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.926 + 0.301i)T + (5.66 + 4.11i)T^{2} \)
17 \( 1 + (2.49 + 1.81i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.89 + 2.23i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + (2.31 - 7.12i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.431 - 0.594i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-10.3 - 3.36i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.44 + 1.11i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + (8.10 - 2.63i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.166 + 0.120i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (9.50 + 3.08i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.822 + 0.597i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.906iT - 67T^{2} \)
71 \( 1 + (-6.72 + 9.25i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.9 + 3.88i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.03 - 4.38i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.34 - 4.60i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 + (-5.49 - 7.56i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.74976699750171230076359489548, −11.53384117748160753702990835934, −10.96230102606571219718783094475, −9.491133964274710365335301656485, −9.191946201098910948729445388020, −7.31024555665727629197519225927, −6.36836848388546240496969408965, −5.03554495980541180127878074184, −3.04542706534999628363238846755, −1.15273399959945115121318117212, 2.76182216093316508925639892911, 4.40789704147063874453890626579, 5.92333818799642845326347853355, 7.29896361146601745879322140518, 7.88244356617587358299438036871, 9.392867957528149083414959863113, 10.11951596582855999964959067664, 11.36517491389804663684290547900, 12.63424353691571343141769126576, 13.10964952351130165766749628656

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