Properties

Label 2-126-3.2-c2-0-0
Degree $2$
Conductor $126$
Sign $-0.816 - 0.577i$
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 − 2.00·4-s + 6.06i·5-s − 2.64·7-s − 2.82i·8-s − 8.58·10-s + 12.1i·11-s − 18.5·13-s − 3.74i·14-s + 4.00·16-s + 10.9i·17-s + 20·19-s − 12.1i·20-s − 17.1·22-s + 12.1i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.21i·5-s − 0.377·7-s − 0.353i·8-s − 0.858·10-s + 1.10i·11-s − 1.42·13-s − 0.267i·14-s + 0.250·16-s + 0.641i·17-s + 1.05·19-s − 0.606i·20-s − 0.780·22-s + 0.527i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.321964 + 1.01298i\)
\(L(\frac12)\) \(\approx\) \(0.321964 + 1.01298i\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + 2.64T \)
good5 \( 1 - 6.06iT - 25T^{2} \)
11 \( 1 - 12.1iT - 121T^{2} \)
13 \( 1 + 18.5T + 169T^{2} \)
17 \( 1 - 10.9iT - 289T^{2} \)
19 \( 1 - 20T + 361T^{2} \)
23 \( 1 - 12.1iT - 529T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 - 25.1T + 961T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 - 60.6iT - 1.68e3T^{2} \)
43 \( 1 - 83.4T + 1.84e3T^{2} \)
47 \( 1 - 16.9iT - 2.20e3T^{2} \)
53 \( 1 + 94.0iT - 2.80e3T^{2} \)
59 \( 1 - 58.2iT - 3.48e3T^{2} \)
61 \( 1 - 15.6T + 3.72e3T^{2} \)
67 \( 1 + 132.T + 4.48e3T^{2} \)
71 \( 1 - 12.1iT - 5.04e3T^{2} \)
73 \( 1 + 76.9T + 5.32e3T^{2} \)
79 \( 1 - 33.6T + 6.24e3T^{2} \)
83 \( 1 - 60.5iT - 6.88e3T^{2} \)
89 \( 1 - 4.77iT - 7.92e3T^{2} \)
97 \( 1 + 188.T + 9.40e3T^{2} \)
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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

−13.77867078853914388086607788300, −12.62638028312899826040281213823, −11.54649006020787776999403142225, −10.07460156197019813110618781554, −9.610827033148991996473449729538, −7.75279603722642839688816168954, −7.10979736704870602959877178938, −5.99217499158942716270492733768, −4.44689333729161806584848471514, −2.73016890208976946518290112092, 0.75400245470374849370968956224, 2.89072373011881194250683301885, 4.57371301637019284148639725453, 5.60445125193025309139142301148, 7.46615070785159274177092263113, 8.813717912415789571405015365226, 9.465806025908761196372359815651, 10.67852178222962961682515393548, 11.96386000464067576160745449485, 12.50448394722767635218986242494

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