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Properties

Label 2-126-63.47-c1-0-7
Degree $2$
Conductor $126$
Sign $0.330 + 0.943i$
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 + (−0.206 − 1.71i)3-s + (0.499 − 0.866i)4-s + 1.42·5-s + (−1.03 − 1.38i)6-s + (−2.43 + 1.02i)7-s − 0.999i·8-s + (−2.91 + 0.708i)9-s + (1.23 − 0.714i)10-s + 3.41i·11-s + (−1.59 − 0.681i)12-s + (5.48 − 3.16i)13-s + (−1.59 + 2.10i)14-s + (−0.294 − 2.45i)15-s + (−0.5 − 0.866i)16-s + (1.14 + 1.97i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.118 − 0.992i)3-s + (0.249 − 0.433i)4-s + 0.639·5-s + (−0.423 − 0.565i)6-s + (−0.921 + 0.388i)7-s − 0.353i·8-s + (−0.971 + 0.236i)9-s + (0.391 − 0.226i)10-s + 1.03i·11-s + (−0.459 − 0.196i)12-s + (1.52 − 0.878i)13-s + (−0.426 + 0.563i)14-s + (−0.0760 − 0.634i)15-s + (−0.125 − 0.216i)16-s + (0.276 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15797 - 0.821487i\)
\(L(\frac12)\) \(\approx\) \(1.15797 - 0.821487i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.206 + 1.71i)T \)
7 \( 1 + (2.43 - 1.02i)T \)
good5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-5.48 + 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.14 - 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.05iT - 23T^{2} \)
29 \( 1 + (0.298 + 0.172i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.76 + 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.202 + 0.350i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.90 + 5.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.75 + 4.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.56 - 4.94i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.51 + 9.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.94 - 5.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 + (0.201 - 0.116i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.811 - 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.02 + 3.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 - 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.12325357122703823438767169848, −12.49569296841914102430083588293, −11.36572724938497377679882935904, −10.20035524730345682412249488671, −9.043671438588170222995887480704, −7.53495967687695027163421629886, −6.22053355449597488859637778725, −5.61265121413073800842518197148, −3.44105790218025346707966533922, −1.85623894669841235587825975888, 3.18316154868688662923641912945, 4.29132311544128849142707085924, 5.88812841749791050315013718498, 6.45707889829808848552986776970, 8.445452658283533619923107691686, 9.390952082548387595220621511502, 10.56835070447272871756849165888, 11.38403897135979962770917701856, 12.80807005495270628516741754744, 13.82767769919277919886290702206

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