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Properties

Label 2-126-63.4-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.190 - 0.981i$
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  + (−0.5 + 0.866i)2-s + (−0.349 + 1.69i)3-s + (−0.499 − 0.866i)4-s + 3.69·5-s + (−1.29 − 1.15i)6-s + (−1.40 + 2.24i)7-s + 0.999·8-s + (−2.75 − 1.18i)9-s + (−1.84 + 3.20i)10-s − 1.47·11-s + (1.64 − 0.545i)12-s + (−1.34 + 2.33i)13-s + (−1.23 − 2.33i)14-s + (−1.29 + 6.27i)15-s + (−0.5 + 0.866i)16-s + (3.28 − 5.69i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.201 + 0.979i)3-s + (−0.249 − 0.433i)4-s + 1.65·5-s + (−0.528 − 0.469i)6-s + (−0.531 + 0.847i)7-s + 0.353·8-s + (−0.918 − 0.395i)9-s + (−0.584 + 1.01i)10-s − 0.445·11-s + (0.474 − 0.157i)12-s + (−0.374 + 0.648i)13-s + (−0.331 − 0.624i)14-s + (−0.334 + 1.62i)15-s + (−0.125 + 0.216i)16-s + (0.797 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613798 + 0.744304i\)
\(L(\frac12)\) \(\approx\) \(0.613798 + 0.744304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.349 - 1.69i)T \)
7 \( 1 + (1.40 - 2.24i)T \)
good5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.28 + 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.444 + 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.28T + 23T^{2} \)
29 \( 1 + (-1.25 - 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.40 + 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.05 - 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.49 + 6.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.60 - 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.45 + 5.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.86 + 4.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.73 - 8.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.72 - 9.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.43 + 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.94766192463054388064119898496, −12.83101482163249803133184390572, −11.37350463318408199483191858968, −10.05968500629716346319034192056, −9.504701238221960355706801512549, −8.846352245948556128457438628791, −6.87986371331111101315767090232, −5.68636237895206756386659361644, −5.08360387653250249905864693145, −2.69230425303319982947513031520, 1.43612196194275166626716796011, 2.96816187328029168737610720753, 5.37424803847900132745699196753, 6.47806916425287717756983414324, 7.67823653668736374535779847882, 8.999889670278148119572426798756, 10.27277851509740775506465707334, 10.65753437091663909238260793522, 12.40714722615674699551365346622, 13.02103497418727532640357225691

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