Properties

Label 2-143-13.12-c3-0-31
Degree $2$
Conductor $143$
Sign $-0.645 + 0.763i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.22i·2-s + 2.89·3-s + 3.05·4-s − 19.1i·5-s − 6.44i·6-s + 4.29i·7-s − 24.5i·8-s − 18.6·9-s − 42.5·10-s + 11i·11-s + 8.86·12-s + (35.7 + 30.2i)13-s + 9.55·14-s − 55.4i·15-s − 30.1·16-s − 7.67·17-s + ⋯
L(s)  = 1  − 0.786i·2-s + 0.557·3-s + 0.382·4-s − 1.71i·5-s − 0.438i·6-s + 0.231i·7-s − 1.08i·8-s − 0.688·9-s − 1.34·10-s + 0.301i·11-s + 0.213·12-s + (0.763 + 0.645i)13-s + 0.182·14-s − 0.954i·15-s − 0.471·16-s − 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.892898 - 1.92391i\)
\(L(\frac12)\) \(\approx\) \(0.892898 - 1.92391i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - 11iT \)
13 \( 1 + (-35.7 - 30.2i)T \)
good2 \( 1 + 2.22iT - 8T^{2} \)
3 \( 1 - 2.89T + 27T^{2} \)
5 \( 1 + 19.1iT - 125T^{2} \)
7 \( 1 - 4.29iT - 343T^{2} \)
17 \( 1 + 7.67T + 4.91e3T^{2} \)
19 \( 1 + 99.3iT - 6.85e3T^{2} \)
23 \( 1 - 15.6T + 1.21e4T^{2} \)
29 \( 1 - 238.T + 2.43e4T^{2} \)
31 \( 1 - 270. iT - 2.97e4T^{2} \)
37 \( 1 + 4.36iT - 5.06e4T^{2} \)
41 \( 1 + 348. iT - 6.89e4T^{2} \)
43 \( 1 + 175.T + 7.95e4T^{2} \)
47 \( 1 + 277. iT - 1.03e5T^{2} \)
53 \( 1 - 714.T + 1.48e5T^{2} \)
59 \( 1 - 831. iT - 2.05e5T^{2} \)
61 \( 1 - 774.T + 2.26e5T^{2} \)
67 \( 1 - 67.2iT - 3.00e5T^{2} \)
71 \( 1 - 534. iT - 3.57e5T^{2} \)
73 \( 1 - 433. iT - 3.89e5T^{2} \)
79 \( 1 + 304.T + 4.93e5T^{2} \)
83 \( 1 - 426. iT - 5.71e5T^{2} \)
89 \( 1 + 100. iT - 7.04e5T^{2} \)
97 \( 1 + 63.8iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.14347586042034618239771621913, −11.56261355450461312088315232807, −10.23463693993474765684138998353, −8.837976433896238096745835580808, −8.704254444678512918264895577984, −6.92469509516899385824175029116, −5.35664273154730005858687493336, −4.00606449781187456890550939951, −2.44246325364849127785773655922, −1.01945339496615248918280764525, 2.49135554064643747239131590840, 3.47366882497540026997911857960, 5.83785658509240032309828912520, 6.51061846385080715747794996412, 7.70175941264856999362433020700, 8.355811809315454338023342991723, 10.07908130542375823319959012516, 10.96993449660497860638523499264, 11.69394477975561433487499152894, 13.51707341097612536476251369782

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