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Properties

Label 2-124-4.3-c2-0-12
Degree $2$
Conductor $124$
Sign $-0.747 - 0.664i$
Analytic cond. $3.37875$
Root an. cond. $1.83813$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.819 + 1.82i)2-s + 3.92i·3-s + (−2.65 + 2.98i)4-s + 7.88·5-s + (−7.16 + 3.21i)6-s + 2.24i·7-s + (−7.63 − 2.40i)8-s − 6.42·9-s + (6.45 + 14.3i)10-s − 17.9i·11-s + (−11.7 − 10.4i)12-s + 7.68·13-s + (−4.10 + 1.84i)14-s + 30.9i·15-s + (−1.86 − 15.8i)16-s − 23.5·17-s + ⋯
L(s)  = 1  + (0.409 + 0.912i)2-s + 1.30i·3-s + (−0.664 + 0.747i)4-s + 1.57·5-s + (−1.19 + 0.536i)6-s + 0.321i·7-s + (−0.953 − 0.300i)8-s − 0.714·9-s + (0.645 + 1.43i)10-s − 1.63i·11-s + (−0.978 − 0.870i)12-s + 0.590·13-s + (−0.293 + 0.131i)14-s + 2.06i·15-s + (−0.116 − 0.993i)16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ -0.747 - 0.664i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.683869 + 1.79801i\)
\(L(\frac12)\) \(\approx\) \(0.683869 + 1.79801i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 1.82i)T \)
31 \( 1 + 5.56iT \)
good3 \( 1 - 3.92iT - 9T^{2} \)
5 \( 1 - 7.88T + 25T^{2} \)
7 \( 1 - 2.24iT - 49T^{2} \)
11 \( 1 + 17.9iT - 121T^{2} \)
13 \( 1 - 7.68T + 169T^{2} \)
17 \( 1 + 23.5T + 289T^{2} \)
19 \( 1 + 8.96iT - 361T^{2} \)
23 \( 1 - 7.13iT - 529T^{2} \)
29 \( 1 - 44.0T + 841T^{2} \)
37 \( 1 + 36.6T + 1.36e3T^{2} \)
41 \( 1 - 4.41T + 1.68e3T^{2} \)
43 \( 1 - 40.4iT - 1.84e3T^{2} \)
47 \( 1 + 7.73iT - 2.20e3T^{2} \)
53 \( 1 + 85.6T + 2.80e3T^{2} \)
59 \( 1 + 73.0iT - 3.48e3T^{2} \)
61 \( 1 + 8.82T + 3.72e3T^{2} \)
67 \( 1 - 94.4iT - 4.48e3T^{2} \)
71 \( 1 - 44.1iT - 5.04e3T^{2} \)
73 \( 1 + 137.T + 5.32e3T^{2} \)
79 \( 1 + 118. iT - 6.24e3T^{2} \)
83 \( 1 + 93.9iT - 6.88e3T^{2} \)
89 \( 1 - 57.5T + 7.92e3T^{2} \)
97 \( 1 + 58.0T + 9.40e3T^{2} \)
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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

−13.72916729372691801527455667437, −13.10928114351905595491210346098, −11.31159198664108403782051273666, −10.25356867228709556047917535818, −9.102515719691896640962444580219, −8.651222966234719650721483181289, −6.46946465033360503794154761703, −5.71185845246253821852400708956, −4.64742514072313790635513454994, −3.09562785585486101573575422602, 1.51601408818767410948756840322, 2.34188182022465032987123751045, 4.63905254843006187195551092672, 6.10965080089285094626375977402, 6.94092793491312197548020615629, 8.728445674907538435312887830695, 9.870049840000355795602444684625, 10.66640237988946878382565148390, 12.18954038075790483396696914655, 12.77068523902415619613716560485

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