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Properties

Label 2-126-21.11-c6-0-14
Degree $2$
Conductor $126$
Sign $-0.434 + 0.900i$
Analytic cond. $28.9868$
Root an. cond. $5.38394$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (168. − 97.0i)5-s + (157. − 304. i)7-s − 181. i·8-s + (548. − 950. i)10-s + (−2.07e3 − 1.20e3i)11-s + 489.·13-s + (−88.5 − 1.93e3i)14-s + (−512. − 886. i)16-s + (4.99e3 + 2.88e3i)17-s + (2.18e3 + 3.77e3i)19-s − 6.20e3i·20-s − 1.35e4·22-s + (−6.40e3 + 3.69e3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.34 − 0.776i)5-s + (0.459 − 0.887i)7-s − 0.353i·8-s + (0.548 − 0.950i)10-s + (−1.56 − 0.901i)11-s + 0.222·13-s + (−0.0322 − 0.706i)14-s + (−0.125 − 0.216i)16-s + (1.01 + 0.587i)17-s + (0.317 + 0.550i)19-s − 0.776i·20-s − 1.27·22-s + (−0.526 + 0.303i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(28.9868\)
Root analytic conductor: \(5.38394\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.388806595\)
\(L(\frac12)\) \(\approx\) \(3.388806595\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.89 + 2.82i)T \)
3 \( 1 \)
7 \( 1 + (-157. + 304. i)T \)
good5 \( 1 + (-168. + 97.0i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (2.07e3 + 1.20e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 489.T + 4.82e6T^{2} \)
17 \( 1 + (-4.99e3 - 2.88e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-2.18e3 - 3.77e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (6.40e3 - 3.69e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 1.85e4iT - 5.94e8T^{2} \)
31 \( 1 + (2.18e4 - 3.77e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (4.44e4 + 7.69e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 7.33e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.07e4T + 6.32e9T^{2} \)
47 \( 1 + (-7.51e4 + 4.33e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-2.22e5 - 1.28e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.84e5 - 1.64e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-3.86e4 - 6.69e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.03e5 - 1.78e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 5.43e5iT - 1.28e11T^{2} \)
73 \( 1 + (3.24e4 - 5.61e4i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-2.22e5 - 3.86e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 4.24e5iT - 3.26e11T^{2} \)
89 \( 1 + (-7.30e5 + 4.21e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 7.45e4T + 8.32e11T^{2} \)
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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

−12.12849163343023402001686128846, −10.54729625740920649914432521169, −10.27703999140826416238538410465, −8.776132833097240353069713411524, −7.55005379172332462743383801545, −5.73176709943939066809235984037, −5.32940471912001821367871465579, −3.70323104260180471541710091407, −2.05980743152964508242876982715, −0.854924219878494617744713953225, 2.05414686787904912430639076178, 2.89334316351414111719981693947, 5.06467443612577746692296381564, 5.67013522593580672751074068251, 6.93381025004302570370738022904, 8.075384124221391230563657651558, 9.581073365723664144059689138558, 10.42279400975216084735961071885, 11.65510253830783136231570695887, 12.81751301049336040640027368373

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