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Properties

Label 2-126-7.2-c9-0-20
Degree $2$
Conductor $126$
Sign $0.438 + 0.898i$
Analytic cond. $64.8945$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 13.8i)2-s + (−127. − 221. i)4-s + (−236. + 409. i)5-s + (−844. + 6.29e3i)7-s + 4.09e3·8-s + (−3.77e3 − 6.54e3i)10-s + (−9.18e3 − 1.59e4i)11-s − 4.75e3·13-s + (−8.04e4 − 6.20e4i)14-s + (−3.27e4 + 5.67e4i)16-s + (−1.31e5 − 2.27e5i)17-s + (−3.58e5 + 6.20e5i)19-s + 1.20e5·20-s + 2.93e5·22-s + (1.29e5 − 2.25e5i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.169 + 0.292i)5-s + (−0.132 + 0.991i)7-s + 0.353·8-s + (−0.119 − 0.207i)10-s + (−0.189 − 0.327i)11-s − 0.0461·13-s + (−0.559 − 0.431i)14-s + (−0.125 + 0.216i)16-s + (−0.381 − 0.660i)17-s + (−0.631 + 1.09i)19-s + 0.169·20-s + 0.267·22-s + (0.0967 − 0.167i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ 0.438 + 0.898i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.3825826834\)
\(L(\frac12)\) \(\approx\) \(0.3825826834\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 13.8i)T \)
3 \( 1 \)
7 \( 1 + (844. - 6.29e3i)T \)
good5 \( 1 + (236. - 409. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (9.18e3 + 1.59e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 4.75e3T + 1.06e10T^{2} \)
17 \( 1 + (1.31e5 + 2.27e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (3.58e5 - 6.20e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-1.29e5 + 2.25e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 4.40e6T + 1.45e13T^{2} \)
31 \( 1 + (-3.01e6 - 5.21e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (-2.28e6 + 3.95e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 1.47e7T + 3.27e14T^{2} \)
43 \( 1 + 2.14e7T + 5.02e14T^{2} \)
47 \( 1 + (-1.90e7 + 3.29e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.58e6 - 2.74e6i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-2.67e7 - 4.64e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-3.06e7 + 5.31e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (1.17e8 + 2.03e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 2.74e8T + 4.58e16T^{2} \)
73 \( 1 + (1.16e8 + 2.02e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.75e7 + 4.77e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 9.49e7T + 1.86e17T^{2} \)
89 \( 1 + (-3.49e8 + 6.05e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.52e9T + 7.60e17T^{2} \)
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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

−11.37122879710958055766272278244, −10.26620479916584402864148927401, −9.118287146983900449784995211809, −8.311245315650488546961241919677, −7.10276157502954314451271134284, −6.02851669809199336090035098671, −5.00433891928677611276875742198, −3.33038758014725548937151344968, −1.88209804986761893724255125947, −0.12555645863656350173035535133, 0.955252321243991270987312943077, 2.34681204870527995328068234946, 3.82855066749489549003060882137, 4.75851657023038262284010513438, 6.55825677890249707797629738105, 7.68441229377767775243515645676, 8.735720665455917620937790595119, 9.870858120483074591027749722177, 10.73568089066266948848652164516, 11.63243751902703957454846187193

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