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Properties

Label 2-124-124.123-c3-0-40
Degree $2$
Conductor $124$
Sign $-0.143 + 0.989i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 2.44i)2-s + 4.58·3-s + (−3.97 − 6.94i)4-s + 8.47·5-s + (6.50 − 11.2i)6-s − 15.2i·7-s + (−22.6 − 0.107i)8-s − 5.93·9-s + (12.0 − 20.7i)10-s + 44.0·11-s + (−18.2 − 31.8i)12-s + 1.16i·13-s + (−37.2 − 21.5i)14-s + 38.8·15-s + (−32.3 + 55.2i)16-s − 38.3i·17-s + ⋯
L(s)  = 1  + (0.501 − 0.865i)2-s + 0.883·3-s + (−0.497 − 0.867i)4-s + 0.757·5-s + (0.442 − 0.764i)6-s − 0.822i·7-s + (−0.999 − 0.00473i)8-s − 0.219·9-s + (0.379 − 0.655i)10-s + 1.20·11-s + (−0.439 − 0.766i)12-s + 0.0249i·13-s + (−0.711 − 0.412i)14-s + 0.669·15-s + (−0.505 + 0.862i)16-s − 0.547i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.143 + 0.989i$
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: \((2,\ 124,\ (\ :3/2),\ -0.143 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.83407 - 2.11972i\)
\(L(\frac12)\) \(\approx\) \(1.83407 - 2.11972i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 2.44i)T \)
31 \( 1 + (135. + 106. i)T \)
good3 \( 1 - 4.58T + 27T^{2} \)
5 \( 1 - 8.47T + 125T^{2} \)
7 \( 1 + 15.2iT - 343T^{2} \)
11 \( 1 - 44.0T + 1.33e3T^{2} \)
13 \( 1 - 1.16iT - 2.19e3T^{2} \)
17 \( 1 + 38.3iT - 4.91e3T^{2} \)
19 \( 1 - 15.7iT - 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 - 302. iT - 2.43e4T^{2} \)
37 \( 1 - 202. iT - 5.06e4T^{2} \)
41 \( 1 - 240.T + 6.89e4T^{2} \)
43 \( 1 - 224.T + 7.95e4T^{2} \)
47 \( 1 + 181. iT - 1.03e5T^{2} \)
53 \( 1 - 376. iT - 1.48e5T^{2} \)
59 \( 1 - 440. iT - 2.05e5T^{2} \)
61 \( 1 - 276. iT - 2.26e5T^{2} \)
67 \( 1 - 220. iT - 3.00e5T^{2} \)
71 \( 1 + 96.4iT - 3.57e5T^{2} \)
73 \( 1 - 119. iT - 3.89e5T^{2} \)
79 \( 1 + 462.T + 4.93e5T^{2} \)
83 \( 1 + 320.T + 5.71e5T^{2} \)
89 \( 1 + 928. iT - 7.04e5T^{2} \)
97 \( 1 - 511.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.87767353054967930449993456786, −11.60665754666415216856274873459, −10.61384016751421398287431158271, −9.444086022375324755372522500245, −8.885391958857782647157307349076, −7.08680979275611152236244515232, −5.65583265540876278131166151975, −4.10962692779301155278525145337, −2.91724591912223238881420142757, −1.37297819273474788598232363985, 2.41531895509473718779818667083, 3.85496758706957612071799249201, 5.52893509830125958091927682430, 6.42654590709606906898751358560, 7.85797342712980758798372233521, 9.001784868542413621349149846563, 9.389977907604895406454852036347, 11.40747711507834969558582216582, 12.55633077347918544601856407443, 13.50352583874238174374298940327

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