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Properties

Label 2-126-21.17-c9-0-7
Degree $2$
Conductor $126$
Sign $-0.0477 - 0.998i$
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  + (−13.8 + 8i)2-s + (127. − 221. i)4-s + (−356. − 618. i)5-s + (3.24e3 + 5.45e3i)7-s + 4.09e3i·8-s + (9.89e3 + 5.71e3i)10-s + (6.38e4 + 3.68e4i)11-s + 5.57e4i·13-s + (−8.86e4 − 4.96e4i)14-s + (−3.27e4 − 5.67e4i)16-s + (1.69e5 − 2.92e5i)17-s + (−3.83e5 + 2.21e5i)19-s − 1.82e5·20-s − 1.17e6·22-s + (−3.35e3 + 1.93e3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.255 − 0.442i)5-s + (0.511 + 0.859i)7-s + 0.353i·8-s + (0.312 + 0.180i)10-s + (1.31 + 0.759i)11-s + 0.541i·13-s + (−0.616 − 0.345i)14-s + (−0.125 − 0.216i)16-s + (0.491 − 0.850i)17-s + (−0.675 + 0.389i)19-s − 0.255·20-s − 1.07·22-s + (−0.00249 + 0.00144i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0477 - 0.998i$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ -0.0477 - 0.998i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.494315225\)
\(L(\frac12)\) \(\approx\) \(1.494315225\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (13.8 - 8i)T \)
3 \( 1 \)
7 \( 1 + (-3.24e3 - 5.45e3i)T \)
good5 \( 1 + (356. + 618. i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (-6.38e4 - 3.68e4i)T + (1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 - 5.57e4iT - 1.06e10T^{2} \)
17 \( 1 + (-1.69e5 + 2.92e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (3.83e5 - 2.21e5i)T + (1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (3.35e3 - 1.93e3i)T + (9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 - 5.37e6iT - 1.45e13T^{2} \)
31 \( 1 + (5.16e6 + 2.98e6i)T + (1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (5.29e6 + 9.17e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 - 2.43e7T + 3.27e14T^{2} \)
43 \( 1 - 2.01e7T + 5.02e14T^{2} \)
47 \( 1 + (7.28e5 + 1.26e6i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (-1.32e7 - 7.64e6i)T + (1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-6.10e7 + 1.05e8i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (1.46e8 - 8.44e7i)T + (5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (2.65e7 - 4.60e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 2.21e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.41e8 + 8.19e7i)T + (2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-3.21e8 - 5.56e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 5.56e8T + 1.86e17T^{2} \)
89 \( 1 + (8.45e6 + 1.46e7i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 - 1.52e9iT - 7.60e17T^{2} \)
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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

−11.92530449831911857043959825735, −10.84714067770458367109058972462, −9.341672947629908264630806370996, −8.936984533504470578169238264415, −7.69085685921855576842021815452, −6.60769239189017022843675018553, −5.33849136185239968272977354571, −4.14141891298560083896770362182, −2.20231331586340518462699285002, −1.09323090694802973490351043661, 0.53198227057995894604677555833, 1.58452962740310399697174781878, 3.26566126555142565436828340175, 4.21035995821218406521971650472, 6.08085101331000348088854073520, 7.23406680250551855439634543545, 8.197736192186943057162239202437, 9.253180038090365323179027298557, 10.56227585118578590541723788811, 11.07722184239936625574998151066

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