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Properties

Label 2-124-124.123-c3-0-8
Degree $2$
Conductor $124$
Sign $-0.985 + 0.171i$
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  + (−0.0971 + 2.82i)2-s + 5.84·3-s + (−7.98 − 0.549i)4-s − 12.9·5-s + (−0.568 + 16.5i)6-s + 20.4i·7-s + (2.32 − 22.5i)8-s + 7.21·9-s + (1.25 − 36.6i)10-s − 52.9·11-s + (−46.6 − 3.21i)12-s + 32.6i·13-s + (−57.7 − 1.98i)14-s − 75.7·15-s + (63.3 + 8.77i)16-s + 52.4i·17-s + ⋯
L(s)  = 1  + (−0.0343 + 0.999i)2-s + 1.12·3-s + (−0.997 − 0.0686i)4-s − 1.15·5-s + (−0.0386 + 1.12i)6-s + 1.10i·7-s + (0.102 − 0.994i)8-s + 0.267·9-s + (0.0398 − 1.15i)10-s − 1.45·11-s + (−1.12 − 0.0773i)12-s + 0.695i·13-s + (−1.10 − 0.0379i)14-s − 1.30·15-s + (0.990 + 0.137i)16-s + 0.747i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.985 + 0.171i$
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: \((2,\ 124,\ (\ :3/2),\ -0.985 + 0.171i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0815574 - 0.945850i\)
\(L(\frac12)\) \(\approx\) \(0.0815574 - 0.945850i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0971 - 2.82i)T \)
31 \( 1 + (-167. + 41.1i)T \)
good3 \( 1 - 5.84T + 27T^{2} \)
5 \( 1 + 12.9T + 125T^{2} \)
7 \( 1 - 20.4iT - 343T^{2} \)
11 \( 1 + 52.9T + 1.33e3T^{2} \)
13 \( 1 - 32.6iT - 2.19e3T^{2} \)
17 \( 1 - 52.4iT - 4.91e3T^{2} \)
19 \( 1 - 1.20iT - 6.85e3T^{2} \)
23 \( 1 - 76.8T + 1.21e4T^{2} \)
29 \( 1 + 66.3iT - 2.43e4T^{2} \)
37 \( 1 - 274. iT - 5.06e4T^{2} \)
41 \( 1 - 236.T + 6.89e4T^{2} \)
43 \( 1 + 72.1T + 7.95e4T^{2} \)
47 \( 1 + 352. iT - 1.03e5T^{2} \)
53 \( 1 - 214. iT - 1.48e5T^{2} \)
59 \( 1 + 64.5iT - 2.05e5T^{2} \)
61 \( 1 - 867. iT - 2.26e5T^{2} \)
67 \( 1 - 134. iT - 3.00e5T^{2} \)
71 \( 1 - 1.00e3iT - 3.57e5T^{2} \)
73 \( 1 - 962. iT - 3.89e5T^{2} \)
79 \( 1 - 462.T + 4.93e5T^{2} \)
83 \( 1 + 1.10e3T + 5.71e5T^{2} \)
89 \( 1 + 1.39e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.30e3T + 9.12e5T^{2} \)
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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

−13.65906516729239296380369979521, −12.76597559419228305307956335733, −11.58219846181444279341035463701, −9.938945315554508932137788826511, −8.619001775145332084772964724494, −8.300257957538970547082332723626, −7.28238057947451190127742262618, −5.68479375987499270730129295768, −4.26092789843482696815052735981, −2.81142000767981542540018492944, 0.43220759895230298219734412101, 2.74695440580956289877793478980, 3.65137571928859490369268195594, 4.91985776208138749283162262060, 7.64643929996787089955122647198, 7.977029745144370689866878491430, 9.248589879671855960934106264553, 10.47141586324831780183223475407, 11.17246860744869163519688019363, 12.50396119448740528872497140720

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