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Properties

Label 2-126-21.20-c3-0-4
Degree $2$
Conductor $126$
Sign $0.907 - 0.419i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
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  + 2i·2-s − 4·4-s + 14.3·5-s + (9.24 − 16.0i)7-s − 8i·8-s + 28.6i·10-s − 59.6i·11-s + 53.7i·13-s + (32.0 + 18.4i)14-s + 16·16-s + 135.·17-s + 82.4i·19-s − 57.2·20-s + 119.·22-s + 20.7i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.28·5-s + (0.499 − 0.866i)7-s − 0.353i·8-s + 0.905i·10-s − 1.63i·11-s + 1.14i·13-s + (0.612 + 0.352i)14-s + 0.250·16-s + 1.93·17-s + 0.994i·19-s − 0.640·20-s + 1.15·22-s + 0.188i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.907 - 0.419i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.95439 + 0.429665i\)
\(L(\frac12)\) \(\approx\) \(1.95439 + 0.429665i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
7 \( 1 + (-9.24 + 16.0i)T \)
good5 \( 1 - 14.3T + 125T^{2} \)
11 \( 1 + 59.6iT - 1.33e3T^{2} \)
13 \( 1 - 53.7iT - 2.19e3T^{2} \)
17 \( 1 - 135.T + 4.91e3T^{2} \)
19 \( 1 - 82.4iT - 6.85e3T^{2} \)
23 \( 1 - 20.7iT - 1.21e4T^{2} \)
29 \( 1 - 63.0iT - 2.43e4T^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + 325.T + 5.06e4T^{2} \)
41 \( 1 + 206.T + 6.89e4T^{2} \)
43 \( 1 - 274.T + 7.95e4T^{2} \)
47 \( 1 - 294.T + 1.03e5T^{2} \)
53 \( 1 + 178. iT - 1.48e5T^{2} \)
59 \( 1 + 170.T + 2.05e5T^{2} \)
61 \( 1 + 72.8iT - 2.26e5T^{2} \)
67 \( 1 + 1.05e3T + 3.00e5T^{2} \)
71 \( 1 - 28.8iT - 3.57e5T^{2} \)
73 \( 1 - 482. iT - 3.89e5T^{2} \)
79 \( 1 + 707.T + 4.93e5T^{2} \)
83 \( 1 - 778.T + 5.71e5T^{2} \)
89 \( 1 + 9.89T + 7.04e5T^{2} \)
97 \( 1 - 1.56e3iT - 9.12e5T^{2} \)
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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

−13.51557000022526407363512985047, −12.01515594125379884532363744057, −10.66127358352460912180754621777, −9.777996415858735544710709489729, −8.640611093664260032422476610342, −7.50505626640792512289213882189, −6.16448870118358628368054161788, −5.38377336391507089430526282673, −3.65918291176385631682580267882, −1.33489200331762259900379488304, 1.61009054929341301007681347288, 2.81843260166762756076719524076, 4.94220174481133008573437369767, 5.74909790558774053811304127927, 7.51091028141900864833352185430, 8.911874156564978945373978323310, 9.881542364351746079633404974125, 10.49852427478598475674533082855, 12.09463474395362620603873126480, 12.57070994993710346520734798870

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