↑

Properties

Label 2-143-13.12-c3-0-26
Degree $2$
Conductor $143$
Sign $0.00908 + 0.999i$
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  + 0.370i·2-s − 0.548·3-s + 7.86·4-s − 11.6i·5-s − 0.203i·6-s − 19.9i·7-s + 5.87i·8-s − 26.6·9-s + 4.30·10-s − 11i·11-s − 4.31·12-s + (−46.8 + 0.425i)13-s + 7.40·14-s + 6.37i·15-s + 60.7·16-s − 6.29·17-s + ⋯
L(s)  = 1  + 0.130i·2-s − 0.105·3-s + 0.982·4-s − 1.03i·5-s − 0.0138i·6-s − 1.07i·7-s + 0.259i·8-s − 0.988·9-s + 0.136·10-s − 0.301i·11-s − 0.103·12-s + (−0.999 + 0.00908i)13-s + 0.141·14-s + 0.109i·15-s + 0.948·16-s − 0.0898·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00908 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00908 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.00908 + 0.999i$
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: \((2,\ 143,\ (\ :3/2),\ 0.00908 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14743 - 1.13705i\)
\(L(\frac12)\) \(\approx\) \(1.14743 - 1.13705i\)
\(L(\frac{5}{2})\) 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 \( 1 + (46.8 - 0.425i)T \)
good2 \( 1 - 0.370iT - 8T^{2} \)
3 \( 1 + 0.548T + 27T^{2} \)
5 \( 1 + 11.6iT - 125T^{2} \)
7 \( 1 + 19.9iT - 343T^{2} \)
17 \( 1 + 6.29T + 4.91e3T^{2} \)
19 \( 1 + 71.8iT - 6.85e3T^{2} \)
23 \( 1 - 37.7T + 1.21e4T^{2} \)
29 \( 1 - 113.T + 2.43e4T^{2} \)
31 \( 1 + 253. iT - 2.97e4T^{2} \)
37 \( 1 - 100. iT - 5.06e4T^{2} \)
41 \( 1 + 25.1iT - 6.89e4T^{2} \)
43 \( 1 - 297.T + 7.95e4T^{2} \)
47 \( 1 - 409. iT - 1.03e5T^{2} \)
53 \( 1 - 385.T + 1.48e5T^{2} \)
59 \( 1 - 416. iT - 2.05e5T^{2} \)
61 \( 1 + 460.T + 2.26e5T^{2} \)
67 \( 1 + 241. iT - 3.00e5T^{2} \)
71 \( 1 - 554. iT - 3.57e5T^{2} \)
73 \( 1 + 402. iT - 3.89e5T^{2} \)
79 \( 1 + 490.T + 4.93e5T^{2} \)
83 \( 1 - 294. iT - 5.71e5T^{2} \)
89 \( 1 + 396. iT - 7.04e5T^{2} \)
97 \( 1 - 852. iT - 9.12e5T^{2} \)
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

−12.28486525467329551498812045062, −11.40371396710444124593299650621, −10.53089446831865366386186228657, −9.218130875470064921215323033947, −8.031891547964644419112024728961, −7.05673750307254688439014506671, −5.78107249821017109816093063395, −4.54150176914239433927172924284, −2.71983249346884732445264091204, −0.77315341055966289484310847492, 2.28154393168294366632922747224, 3.10954874464457767220226045881, 5.39283640130029932114025429386, 6.41736213671770990140245940878, 7.38834177547385662921352289180, 8.698012534051886553331235915438, 10.10095593194964835535877053073, 10.92869919306155374864437245477, 11.91922876489439602954549534525, 12.41457845220556826726385394538

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