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Properties

Label 2-126-63.4-c1-0-4
Degree $2$
Conductor $126$
Sign $0.945 + 0.325i$
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 + (1.29 + 1.15i)3-s + (−0.499 − 0.866i)4-s + 1.58·5-s + (1.64 − 0.545i)6-s + (−2.64 − 0.0963i)7-s − 0.999·8-s + (0.349 + 2.97i)9-s + (0.794 − 1.37i)10-s − 1.58·11-s + (0.349 − 1.69i)12-s + (2.40 − 4.16i)13-s + (−1.40 + 2.24i)14-s + (2.05 + 1.82i)15-s + (−0.5 + 0.866i)16-s + (−2.69 + 4.67i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.747 + 0.664i)3-s + (−0.249 − 0.433i)4-s + 0.710·5-s + (0.671 − 0.222i)6-s + (−0.999 − 0.0364i)7-s − 0.353·8-s + (0.116 + 0.993i)9-s + (0.251 − 0.434i)10-s − 0.478·11-s + (0.100 − 0.489i)12-s + (0.667 − 1.15i)13-s + (−0.375 + 0.599i)14-s + (0.530 + 0.472i)15-s + (−0.125 + 0.216i)16-s + (−0.654 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.945 + 0.325i$
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: \((2,\ 126,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47136 - 0.246514i\)
\(L(\frac12)\) \(\approx\) \(1.47136 - 0.246514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.15i)T \)
7 \( 1 + (2.64 + 0.0963i)T \)
good5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.300T + 23T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.35 - 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 - 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.23 + 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.14718379645617777027939674410, −12.82696384527202456306232450786, −10.80959374375474514306447278755, −10.38113423182451755012943110806, −9.290421320797314516858495555344, −8.383211945599749672592081385757, −6.49026569704850197614274030391, −5.17163324064884639422067900811, −3.65973002710619229074729051696, −2.48978995753422717894583952160, 2.43302439274686321422683307550, 4.03751028382164272898075285560, 6.04174285103951081141132427393, 6.66515354680586032019837820974, 7.974231710209462604403918772888, 9.111624442749112169553934453857, 9.915607757485659977302428281154, 11.74977256690892120164606014717, 12.84057113014324673595875578599, 13.60823125614177564423194071656

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