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Properties

Label 2-126-63.11-c2-0-7
Degree $2$
Conductor $126$
Sign $0.899 + 0.437i$
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  − 1.41i·2-s + (2.36 + 1.84i)3-s − 2.00·4-s + (1.22 + 0.707i)5-s + (2.61 − 3.34i)6-s + (3.10 − 6.27i)7-s + 2.82i·8-s + (2.17 + 8.73i)9-s + (1.00 − 1.73i)10-s + (15.1 − 8.72i)11-s + (−4.72 − 3.69i)12-s + (8.59 + 14.8i)13-s + (−8.87 − 4.38i)14-s + (1.58 + 3.93i)15-s + 4.00·16-s + (−20.8 − 12.0i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.788 + 0.615i)3-s − 0.500·4-s + (0.244 + 0.141i)5-s + (0.435 − 0.557i)6-s + (0.443 − 0.896i)7-s + 0.353i·8-s + (0.241 + 0.970i)9-s + (0.100 − 0.173i)10-s + (1.37 − 0.793i)11-s + (−0.394 − 0.307i)12-s + (0.660 + 1.14i)13-s + (−0.633 − 0.313i)14-s + (0.105 + 0.262i)15-s + 0.250·16-s + (−1.22 − 0.708i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.899 + 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.80779 - 0.416279i\)
\(L(\frac12)\) \(\approx\) \(1.80779 - 0.416279i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-2.36 - 1.84i)T \)
7 \( 1 + (-3.10 + 6.27i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-15.1 + 8.72i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.59 - 14.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (20.8 + 12.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.7 - 20.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (27.1 + 15.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13.7 + 7.94i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 32.1T + 961T^{2} \)
37 \( 1 + (-15.7 - 27.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-8.67 + 5.00i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (31.6 - 54.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 4.27iT - 2.20e3T^{2} \)
53 \( 1 + (45.3 + 26.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 42.1iT - 3.48e3T^{2} \)
61 \( 1 + 12.4T + 3.72e3T^{2} \)
67 \( 1 - 96.1T + 4.48e3T^{2} \)
71 \( 1 + 20.8iT - 5.04e3T^{2} \)
73 \( 1 + (-41.6 + 72.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 45.1T + 6.24e3T^{2} \)
83 \( 1 + (-3.73 - 2.15i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-123. + 71.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (17.5 - 30.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.41798275944876583434430125120, −11.69703856616252849497243496161, −11.03469116953065777256246533538, −9.880460931521452262352066650042, −9.064938153724245469961299327023, −8.032300271744783177684855630740, −6.41466609326211054325451714003, −4.41269918100743273689869533840, −3.67837941461927612226489632340, −1.78505693148546906156090459139, 1.82457168331083138797884389799, 3.81907786366031975260324231777, 5.56790695286681668870777722677, 6.70695401993968504456701714332, 7.84421435522950740900840302006, 8.900667220953553962607009615543, 9.474154205303653611291041605238, 11.34595436614626427118904226095, 12.52563837140227144079292388980, 13.30423985369621039130425047068

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