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Properties

Label 2-126-21.5-c1-0-2
Degree $2$
Conductor $126$
Sign $0.811 - 0.584i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.358 + 0.621i)5-s + (2.62 − 0.358i)7-s + 0.999i·8-s + (−0.621 + 0.358i)10-s + (−2.59 + 1.5i)11-s − 2.44i·13-s + (2.44 + i)14-s + (−0.5 + 0.866i)16-s + (−2.95 − 5.12i)17-s + (−5.12 − 2.95i)19-s − 0.717·20-s − 3·22-s + (3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.160 + 0.277i)5-s + (0.990 − 0.135i)7-s + 0.353i·8-s + (−0.196 + 0.113i)10-s + (−0.783 + 0.452i)11-s − 0.679i·13-s + (0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.717 − 1.24i)17-s + (−1.17 − 0.678i)19-s − 0.160·20-s − 0.639·22-s + (0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38531 + 0.447369i\)
\(L(\frac12)\) \(\approx\) \(1.38531 + 0.447369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (0.358 - 0.621i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 + (2.95 - 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.27 - 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.03 - 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 - 0.717i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−13.44450075468445021650637676417, −12.71668343583440377573040993958, −11.33163303082220364321529356939, −10.77833352808826150874196431706, −9.115557876076186430712062334642, −7.81382965727956037778473452361, −7.03506904324746926597308134117, −5.41334688975840668252728547195, −4.45193754230990458553957168598, −2.62211231102784034099887044275, 2.05834620492555465827604442691, 4.02626666096715424137667738268, 5.12498201823083178854217955983, 6.43627401669092822679075538678, 8.020652442567226168828577081785, 8.941490456646942770409300722465, 10.69332231544337470716242049262, 11.07846504041332774955743055502, 12.46804930934061774988690175654, 13.04973586264021410010624709693

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