↑

Properties

Label 2-124-124.123-c3-0-34
Degree $2$
Conductor $124$
Sign $0.544 - 0.838i$
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  + (2.09 + 1.89i)2-s + 10.0·3-s + (0.807 + 7.95i)4-s − 5.59·5-s + (21.1 + 19.1i)6-s − 15.9i·7-s + (−13.4 + 18.2i)8-s + 74.5·9-s + (−11.7 − 10.6i)10-s − 22.3·11-s + (8.13 + 80.2i)12-s + 62.3i·13-s + (30.2 − 33.4i)14-s − 56.4·15-s + (−62.6 + 12.8i)16-s − 61.2i·17-s + ⋯
L(s)  = 1  + (0.741 + 0.670i)2-s + 1.93·3-s + (0.100 + 0.994i)4-s − 0.500·5-s + (1.43 + 1.30i)6-s − 0.860i·7-s + (−0.592 + 0.805i)8-s + 2.76·9-s + (−0.371 − 0.335i)10-s − 0.613·11-s + (0.195 + 1.92i)12-s + 1.32i·13-s + (0.576 − 0.638i)14-s − 0.971·15-s + (−0.979 + 0.200i)16-s − 0.874i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.544 - 0.838i$
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: \((2,\ 124,\ (\ :3/2),\ 0.544 - 0.838i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.37226 + 1.83119i\)
\(L(\frac12)\) \(\approx\) \(3.37226 + 1.83119i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.09 - 1.89i)T \)
31 \( 1 + (134. + 108. i)T \)
good3 \( 1 - 10.0T + 27T^{2} \)
5 \( 1 + 5.59T + 125T^{2} \)
7 \( 1 + 15.9iT - 343T^{2} \)
11 \( 1 + 22.3T + 1.33e3T^{2} \)
13 \( 1 - 62.3iT - 2.19e3T^{2} \)
17 \( 1 + 61.2iT - 4.91e3T^{2} \)
19 \( 1 + 137. iT - 6.85e3T^{2} \)
23 \( 1 + 30.7T + 1.21e4T^{2} \)
29 \( 1 - 211. iT - 2.43e4T^{2} \)
37 \( 1 + 115. iT - 5.06e4T^{2} \)
41 \( 1 + 15.6T + 6.89e4T^{2} \)
43 \( 1 - 280.T + 7.95e4T^{2} \)
47 \( 1 - 1.51iT - 1.03e5T^{2} \)
53 \( 1 - 214. iT - 1.48e5T^{2} \)
59 \( 1 + 494. iT - 2.05e5T^{2} \)
61 \( 1 - 359. iT - 2.26e5T^{2} \)
67 \( 1 - 452. iT - 3.00e5T^{2} \)
71 \( 1 - 152. iT - 3.57e5T^{2} \)
73 \( 1 + 45.8iT - 3.89e5T^{2} \)
79 \( 1 + 42.6T + 4.93e5T^{2} \)
83 \( 1 + 320.T + 5.71e5T^{2} \)
89 \( 1 - 859. iT - 7.04e5T^{2} \)
97 \( 1 + 763.T + 9.12e5T^{2} \)
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

−13.60658741155599006248121536747, −12.61760366516317202794800419330, −11.14910320542050848854195918028, −9.461390561262334528129837055189, −8.649076168530195748468888956534, −7.39480273024767800460058763568, −7.12447812008464029202580644843, −4.60836361588632031804538822325, −3.75242942270468553969098516576, −2.47387549696507597301952559897, 1.95152443382523049608249426134, 3.10899453625133393255092182541, 4.04333136062760099905524674986, 5.77323775808495515085284615561, 7.73478520110696610706198190392, 8.427285640734775621016769857797, 9.722751330274536107009652651395, 10.48711193083339483555875133677, 12.22895512416248488738010268193, 12.82472052219616784058529209089

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