↑

Properties

Label 2-124-31.20-c1-0-1
Degree $2$
Conductor $124$
Sign $0.700 - 0.714i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.25 + 2.50i)3-s + (−0.443 − 0.767i)5-s + (−0.381 − 3.63i)7-s + (−0.874 + 8.32i)9-s + (−3.63 − 1.61i)11-s + (−1.31 + 0.278i)13-s + (0.923 − 2.84i)15-s + (0.825 − 0.367i)17-s + (3.79 + 0.807i)19-s + (8.23 − 9.14i)21-s + (−1.81 − 1.32i)23-s + (2.10 − 3.64i)25-s + (−14.6 + 10.6i)27-s + (2.23 + 6.87i)29-s + (−3.89 − 3.97i)31-s + ⋯
L(s)  = 1  + (1.30 + 1.44i)3-s + (−0.198 − 0.343i)5-s + (−0.144 − 1.37i)7-s + (−0.291 + 2.77i)9-s + (−1.09 − 0.487i)11-s + (−0.363 + 0.0772i)13-s + (0.238 − 0.733i)15-s + (0.200 − 0.0891i)17-s + (0.871 + 0.185i)19-s + (1.79 − 1.99i)21-s + (−0.379 − 0.275i)23-s + (0.421 − 0.729i)25-s + (−2.81 + 2.04i)27-s + (0.414 + 1.27i)29-s + (−0.699 − 0.714i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.700 - 0.714i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.700 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28923 + 0.541443i\)
\(L(\frac12)\) \(\approx\) \(1.28923 + 0.541443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (3.89 + 3.97i)T \)
good3 \( 1 + (-2.25 - 2.50i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (0.443 + 0.767i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.381 + 3.63i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (3.63 + 1.61i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (1.31 - 0.278i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-0.825 + 0.367i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (-3.79 - 0.807i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (1.81 + 1.32i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.23 - 6.87i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (-0.608 + 1.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.12 - 3.47i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-5.73 - 1.21i)T + (39.2 + 17.4i)T^{2} \)
47 \( 1 + (0.483 - 1.48i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.01 - 9.61i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (7.47 + 8.30i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 - 9.66T + 61T^{2} \)
67 \( 1 + (0.0443 + 0.0768i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.513 + 4.88i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (-6.21 - 2.76i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (5.54 - 2.46i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (4.51 - 5.01i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (5.56 - 4.04i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.08 + 0.787i)T + (29.9 - 92.2i)T^{2} \)
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

−13.91488727505058390308862818054, −12.86333589317790747224960891919, −10.96046395600084638063394074127, −10.30420170322775453593019683394, −9.454071473433549035066618032574, −8.262531503629589117317637392272, −7.50530102532779504124490505004, −5.12330518920860858580887157307, −4.09861753057884071543049755569, −2.96755085968932612415033758150, 2.19383492356694149314724537235, 3.13376804442329349808821223878, 5.63433267682866983161765011068, 7.03743044531334512085050331853, 7.85003745813949676407998726771, 8.793690148816503158746017359568, 9.794616920328975597450394700376, 11.70196618979584188184415735320, 12.46833923794387319316464746095, 13.20135220601989566706040130218

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