Properties

Label 2-126-7.5-c2-0-1
Degree $2$
Conductor $126$
Sign $0.605 - 0.795i$
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  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (4.24 + 2.44i)5-s + (3.5 − 6.06i)7-s + 2.82·8-s + (−6 + 3.46i)10-s + (8.48 + 14.6i)11-s + 1.73i·13-s + (4.94 + 8.57i)14-s + (−2.00 + 3.46i)16-s + (−4.24 + 2.44i)17-s + (25.5 + 14.7i)19-s − 9.79i·20-s − 24·22-s + (4.24 − 7.34i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.848 + 0.489i)5-s + (0.5 − 0.866i)7-s + 0.353·8-s + (−0.600 + 0.346i)10-s + (0.771 + 1.33i)11-s + 0.133i·13-s + (0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (−0.249 + 0.144i)17-s + (1.34 + 0.774i)19-s − 0.489i·20-s − 1.09·22-s + (0.184 − 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24518 + 0.617269i\)
\(L(\frac12)\) \(\approx\) \(1.24518 + 0.617269i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (-4.24 - 2.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.48 - 14.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.73iT - 169T^{2} \)
17 \( 1 + (4.24 - 2.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.5 - 14.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-4.24 + 7.34i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (10.5 - 6.06i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-23.5 + 40.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 68.5iT - 1.68e3T^{2} \)
43 \( 1 - 31T + 1.84e3T^{2} \)
47 \( 1 + (72.1 + 41.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-38.1 - 66.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (72.1 - 41.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (72 + 41.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 59.3T + 5.04e3T^{2} \)
73 \( 1 + (70.5 - 40.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.5 - 35.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 4.89iT - 6.88e3T^{2} \)
89 \( 1 + (-50.9 - 29.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 41.5iT - 9.40e3T^{2} \)
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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

−13.67506276566132299886569075531, −12.31818648314265576802948045762, −10.94341219963813587388884363205, −9.985874771939048542667734139859, −9.228615623349509868734665926540, −7.61111499328648994569052258525, −6.90634797381019122175765268389, −5.60307507354846873277134514169, −4.13883948716215062072365424699, −1.72949438704184250308697401804, 1.40164352030399668377685839969, 3.09189817563065502634708153849, 5.02746143245630392923879889175, 6.11240567253940469029673690585, 7.931782535984101302990678554889, 9.104983254988808411901091064404, 9.520453661739833870504139055403, 11.21623596621792018733500585365, 11.64464992913330029569511211739, 13.03407484927168163470907300496

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