↑

Properties

Label 2-126-63.41-c1-0-6
Degree $2$
Conductor $126$
Sign $0.274 + 0.961i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.69 − 0.354i)3-s + (0.499 − 0.866i)4-s + (0.895 − 1.55i)5-s + (−1.64 + 0.541i)6-s + (0.0213 − 2.64i)7-s − 0.999i·8-s + (2.74 + 1.20i)9-s − 1.79i·10-s + (−2.07 + 1.20i)11-s + (−1.15 + 1.29i)12-s + (4.23 + 2.44i)13-s + (−1.30 − 2.30i)14-s + (−2.06 + 2.31i)15-s + (−0.5 − 0.866i)16-s − 3.66·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.978 − 0.204i)3-s + (0.249 − 0.433i)4-s + (0.400 − 0.693i)5-s + (−0.671 + 0.220i)6-s + (0.00808 − 0.999i)7-s − 0.353i·8-s + (0.916 + 0.400i)9-s − 0.566i·10-s + (−0.627 + 0.362i)11-s + (−0.333 + 0.372i)12-s + (1.17 + 0.678i)13-s + (−0.348 − 0.615i)14-s + (−0.533 + 0.596i)15-s + (−0.125 − 0.216i)16-s − 0.888·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.274 + 0.961i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.274 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939864 - 0.708974i\)
\(L(\frac12)\) \(\approx\) \(0.939864 - 0.708974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.354i)T \)
7 \( 1 + (-0.0213 + 2.64i)T \)
good5 \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.23 - 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.68 - 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.02 - 2.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (-4.04 + 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.48 + 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.56 + 4.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (7.29 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.81 - 5.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.285 - 0.493i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + (4.77 - 2.75i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.33512660034726510034812164826, −12.24584553398189335249470567135, −11.09256994518741068755221942737, −10.49711804619812114170703260587, −9.173870086734967143692917921343, −7.42749663162572500354476235396, −6.28841314964516166251429124786, −5.12061754096940478467678961621, −4.06119041334726301298038479079, −1.46060520192231330092025598193, 2.84053991368194812539677828191, 4.69395411285464714672551997765, 5.94870867863340291300382403230, 6.44318936078177446392914099822, 8.072704272573568076385703857959, 9.512557021306789697323279670230, 10.98158330958519942370867823452, 11.30154759613966343960088802925, 12.80828794153723007879119964824, 13.31750454885976851892779711794

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