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Properties

Label 2-126-63.2-c2-0-9
Degree $2$
Conductor $126$
Sign $0.156 + 0.987i$
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  + (−1.22 + 0.707i)2-s + (−0.515 + 2.95i)3-s + (0.999 − 1.73i)4-s − 7.66i·5-s + (−1.45 − 3.98i)6-s + (−6.97 + 0.633i)7-s + 2.82i·8-s + (−8.46 − 3.04i)9-s + (5.42 + 9.38i)10-s − 9.96i·11-s + (4.60 + 3.84i)12-s + (−5.46 − 9.46i)13-s + (8.08 − 5.70i)14-s + (22.6 + 3.95i)15-s + (−2.00 − 3.46i)16-s + (24.5 − 14.1i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.171 + 0.985i)3-s + (0.249 − 0.433i)4-s − 1.53i·5-s + (−0.242 − 0.664i)6-s + (−0.995 + 0.0905i)7-s + 0.353i·8-s + (−0.940 − 0.338i)9-s + (0.542 + 0.938i)10-s − 0.905i·11-s + (0.383 + 0.320i)12-s + (−0.420 − 0.727i)13-s + (0.577 − 0.407i)14-s + (1.51 + 0.263i)15-s + (−0.125 − 0.216i)16-s + (1.44 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.421360 - 0.359728i\)
\(L(\frac12)\) \(\approx\) \(0.421360 - 0.359728i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (0.515 - 2.95i)T \)
7 \( 1 + (6.97 - 0.633i)T \)
good5 \( 1 + 7.66iT - 25T^{2} \)
11 \( 1 + 9.96iT - 121T^{2} \)
13 \( 1 + (5.46 + 9.46i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-24.5 + 14.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (5.29 - 9.17i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 0.743iT - 529T^{2} \)
29 \( 1 + (39.7 + 22.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.8 - 41.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-12.6 + 21.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (32.2 - 18.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.0 + 19.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-25.8 + 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (26.8 - 15.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.6 + 29.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-12.8 - 22.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.804 - 1.39i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 98.6iT - 5.04e3T^{2} \)
73 \( 1 + (-43.7 - 75.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (53.6 + 93.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-135. - 78.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-54.2 - 31.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-30.3 + 52.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.76493528879273804842635034042, −11.87165213964569613347725543719, −10.47672839384483433300153602053, −9.539586888061641113650035220106, −8.927222458782770165901701671832, −7.81252347068695638656754559256, −5.87035959743576940803357190793, −5.18592319876808824789636870049, −3.45329453267397827535268145830, −0.45775723104326996610814426520, 2.13095864307079351451435102055, 3.41927753051595259906485496871, 6.08385886639459607557176291850, 7.03541402351733283745812411179, 7.62897345449098606681556019577, 9.364170115076486564681183456596, 10.30777685286756917202590216764, 11.26283343344737148258410277620, 12.28549736066552683471348585919, 13.10789772522556851446161398061

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