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Properties

Label 2-126-7.5-c2-0-4
Degree $2$
Conductor $126$
Sign $-0.605 + 0.795i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.24 − 2.44i)5-s + (3.5 − 6.06i)7-s − 2.82·8-s + (−6 + 3.46i)10-s + (−8.48 − 14.6i)11-s + 1.73i·13-s + (−4.94 − 8.57i)14-s + (−2.00 + 3.46i)16-s + (4.24 − 2.44i)17-s + (25.5 + 14.7i)19-s + 9.79i·20-s − 24·22-s + (−4.24 + 7.34i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.848 − 0.489i)5-s + (0.5 − 0.866i)7-s − 0.353·8-s + (−0.600 + 0.346i)10-s + (−0.771 − 1.33i)11-s + 0.133i·13-s + (−0.353 − 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.249 − 0.144i)17-s + (1.34 + 0.774i)19-s + 0.489i·20-s − 1.09·22-s + (−0.184 + 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.581636 - 1.17329i\)
\(L(\frac12)\) \(\approx\) \(0.581636 - 1.17329i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (4.24 + 2.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.48 + 14.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.73iT - 169T^{2} \)
17 \( 1 + (-4.24 + 2.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.5 - 14.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.24 - 7.34i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (10.5 - 6.06i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-23.5 + 40.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 68.5iT - 1.68e3T^{2} \)
43 \( 1 - 31T + 1.84e3T^{2} \)
47 \( 1 + (-72.1 - 41.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.1 + 66.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-72.1 + 41.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (72 + 41.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 59.3T + 5.04e3T^{2} \)
73 \( 1 + (70.5 - 40.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.5 - 35.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 4.89iT - 6.88e3T^{2} \)
89 \( 1 + (50.9 + 29.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 41.5iT - 9.40e3T^{2} \)
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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

−12.70705953654694493213680996179, −11.64748917440166069847306199961, −10.98380081129174727720323264200, −9.861491066926280042594366057378, −8.374140411513708548235747319416, −7.58828466632123859488811935787, −5.71278772400842597911187398348, −4.45354982415260915040560389120, −3.26997428528081853066328780507, −0.865737985357350770169734339365, 2.76280684712759249749233470435, 4.46962888631178437753923665646, 5.56683627772490433003093989572, 7.17300109856377330190283838977, 7.81328367083374943263496844457, 9.077109939226296806601124150937, 10.44823797998706838696595742595, 11.78160157882963505100592081609, 12.33904416409889263424382742899, 13.62487310458980736451019919914

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