↑

Properties

Label 2-124-124.123-c3-0-30
Degree $2$
Conductor $124$
Sign $0.646 + 0.763i$
Analytic cond. $7.31623$
Root an. cond. $2.70485$
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  + (2.57 − 1.16i)2-s − 4.19·3-s + (5.27 − 6.01i)4-s + 15.8·5-s + (−10.8 + 4.89i)6-s + 12.2i·7-s + (6.59 − 21.6i)8-s − 9.40·9-s + (40.7 − 18.4i)10-s + 58.7·11-s + (−22.1 + 25.2i)12-s − 87.1i·13-s + (14.2 + 31.5i)14-s − 66.4·15-s + (−8.24 − 63.4i)16-s + 90.2i·17-s + ⋯
L(s)  = 1  + (0.911 − 0.412i)2-s − 0.807·3-s + (0.659 − 0.751i)4-s + 1.41·5-s + (−0.735 + 0.332i)6-s + 0.661i·7-s + (0.291 − 0.956i)8-s − 0.348·9-s + (1.29 − 0.583i)10-s + 1.61·11-s + (−0.532 + 0.606i)12-s − 1.85i·13-s + (0.272 + 0.602i)14-s − 1.14·15-s + (−0.128 − 0.991i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.646 + 0.763i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.646 + 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.42841 - 1.12552i\)
\(L(\frac12)\) \(\approx\) \(2.42841 - 1.12552i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.57 + 1.16i)T \)
31 \( 1 + (25.3 - 170. i)T \)
good3 \( 1 + 4.19T + 27T^{2} \)
5 \( 1 - 15.8T + 125T^{2} \)
7 \( 1 - 12.2iT - 343T^{2} \)
11 \( 1 - 58.7T + 1.33e3T^{2} \)
13 \( 1 + 87.1iT - 2.19e3T^{2} \)
17 \( 1 - 90.2iT - 4.91e3T^{2} \)
19 \( 1 + 67.1iT - 6.85e3T^{2} \)
23 \( 1 + 29.5T + 1.21e4T^{2} \)
29 \( 1 - 51.5iT - 2.43e4T^{2} \)
37 \( 1 - 181. iT - 5.06e4T^{2} \)
41 \( 1 + 356.T + 6.89e4T^{2} \)
43 \( 1 + 430.T + 7.95e4T^{2} \)
47 \( 1 - 427. iT - 1.03e5T^{2} \)
53 \( 1 - 441. iT - 1.48e5T^{2} \)
59 \( 1 - 483. iT - 2.05e5T^{2} \)
61 \( 1 + 590. iT - 2.26e5T^{2} \)
67 \( 1 + 236. iT - 3.00e5T^{2} \)
71 \( 1 + 281. iT - 3.57e5T^{2} \)
73 \( 1 + 53.2iT - 3.89e5T^{2} \)
79 \( 1 + 637.T + 4.93e5T^{2} \)
83 \( 1 - 111.T + 5.71e5T^{2} \)
89 \( 1 + 226. iT - 7.04e5T^{2} \)
97 \( 1 + 67.1T + 9.12e5T^{2} \)
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

−12.69467360177814692544926982718, −11.95441643434751251785954385292, −10.82101039208351690363061760877, −10.04265460660270857094889336522, −8.774779161128129633805852839003, −6.48989330922220187311678160666, −5.92024067004225196976081964574, −5.06322827315983988231217523459, −3.08900494159608288591540376691, −1.44144665032478132683614786758, 1.86961755541354395466070290245, 3.98646701635326085947152199448, 5.28367441644312822737093061800, 6.38129810095741464316430790982, 6.89511375625293139271577513544, 8.915093508109704451578337773317, 10.00650772579903627119798868680, 11.60358549816600997886001538485, 11.77196992538040037215262030588, 13.40660392939161591806211306273

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