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Properties

Label 2-126-63.20-c3-0-23
Degree $2$
Conductor $126$
Sign $-0.765 - 0.643i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (2.25 − 4.68i)3-s + (1.99 + 3.46i)4-s + (−7.47 − 12.9i)5-s + (−8.58 + 5.84i)6-s + (−11.5 + 14.4i)7-s − 7.99i·8-s + (−16.8 − 21.1i)9-s + 29.9i·10-s + (3.00 + 1.73i)11-s + (20.7 − 1.54i)12-s + (−40.9 + 23.6i)13-s + (34.4 − 13.5i)14-s + (−77.4 + 5.75i)15-s + (−8 + 13.8i)16-s + 90.3·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.434 − 0.900i)3-s + (0.249 + 0.433i)4-s + (−0.668 − 1.15i)5-s + (−0.584 + 0.397i)6-s + (−0.622 + 0.782i)7-s − 0.353i·8-s + (−0.622 − 0.782i)9-s + 0.945i·10-s + (0.0824 + 0.0476i)11-s + (0.498 − 0.0370i)12-s + (−0.872 + 0.503i)13-s + (0.658 − 0.258i)14-s + (−1.33 + 0.0991i)15-s + (−0.125 + 0.216i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.765 - 0.643i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.765 - 0.643i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.133075 + 0.365390i\)
\(L(\frac12)\) \(\approx\) \(0.133075 + 0.365390i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 + (-2.25 + 4.68i)T \)
7 \( 1 + (11.5 - 14.4i)T \)
good5 \( 1 + (7.47 + 12.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-3.00 - 1.73i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (40.9 - 23.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 90.3T + 4.91e3T^{2} \)
19 \( 1 + 43.3iT - 6.85e3T^{2} \)
23 \( 1 + (182. - 105. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (51.5 + 29.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (155. - 89.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 73.6T + 5.06e4T^{2} \)
41 \( 1 + (54.2 + 94.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-117. + 203. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-215. + 374. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 466. iT - 1.48e5T^{2} \)
59 \( 1 + (166. + 288. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (247. + 143. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-461. - 799. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 852. iT - 3.57e5T^{2} \)
73 \( 1 + 478. iT - 3.89e5T^{2} \)
79 \( 1 + (448. - 777. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-417. + 722. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 + (384. + 221. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.14502292314357194179566664865, −11.79530636356776600184055119983, −9.787968852397022785245408230000, −8.991671808792537654308243015073, −8.103786255894245915503406066974, −7.14023531033929083450732089082, −5.54245267940601439800281449745, −3.59372648766341663397821428546, −1.93693346360763796352718739712, −0.22192285199762388001509478905, 2.91262939240136080849262575132, 4.05643022328380137412633580652, 5.91142417229703003809992782705, 7.37419981573383596203081032496, 7.981792696069058962001605321542, 9.610613291008360594102649585267, 10.25820469291714987856930893972, 10.96793268164798736081723079328, 12.34175812715827342170680832008, 14.12805017978437076924265594892

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