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Properties

Label 2-126-63.58-c3-0-20
Degree $2$
Conductor $126$
Sign $0.209 + 0.977i$
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  + 2·2-s + (2.30 − 4.65i)3-s + 4·4-s + (−5.06 − 8.77i)5-s + (4.61 − 9.31i)6-s + (14.4 + 11.5i)7-s + 8·8-s + (−16.3 − 21.4i)9-s + (−10.1 − 17.5i)10-s + (19.0 − 33.0i)11-s + (9.23 − 18.6i)12-s + (−1.46 + 2.53i)13-s + (28.8 + 23.1i)14-s + (−52.5 + 3.32i)15-s + 16·16-s + (−56.0 − 97.0i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.444 − 0.895i)3-s + 0.5·4-s + (−0.453 − 0.785i)5-s + (0.314 − 0.633i)6-s + (0.779 + 0.626i)7-s + 0.353·8-s + (−0.605 − 0.796i)9-s + (−0.320 − 0.555i)10-s + (0.522 − 0.904i)11-s + (0.222 − 0.447i)12-s + (−0.0311 + 0.0539i)13-s + (0.551 + 0.442i)14-s + (−0.904 + 0.0572i)15-s + 0.250·16-s + (−0.799 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.12510 - 1.71766i\)
\(L(\frac12)\) \(\approx\) \(2.12510 - 1.71766i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (-2.30 + 4.65i)T \)
7 \( 1 + (-14.4 - 11.5i)T \)
good5 \( 1 + (5.06 + 8.77i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-19.0 + 33.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (1.46 - 2.53i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (56.0 + 97.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.2 - 68.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-73.5 - 127. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-137. - 238. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 102.T + 2.97e4T^{2} \)
37 \( 1 + (25.1 - 43.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-1.16 + 2.01i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-184. - 318. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 399.T + 1.03e5T^{2} \)
53 \( 1 + (50.8 + 88.0i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 27.4T + 2.05e5T^{2} \)
61 \( 1 + 612.T + 2.26e5T^{2} \)
67 \( 1 + 624.T + 3.00e5T^{2} \)
71 \( 1 + 514.T + 3.57e5T^{2} \)
73 \( 1 + (-590. - 1.02e3i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 340.T + 4.93e5T^{2} \)
83 \( 1 + (312. + 540. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-349. + 604. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (345. + 599. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.63330140447982200407780502147, −11.90515623603996955836115845198, −11.18719710369460128131785716072, −9.051389340530947258695228023608, −8.381449461355983607791918750070, −7.19840263317027428863634316311, −5.87947189023992653741034385441, −4.61401088363771592724799902703, −2.95180391331499559523707992507, −1.25677990130994871382087673434, 2.41994715414349934713789381419, 4.02761209105613473510815136957, 4.62799923248275902019059571694, 6.48839995440703978264046138928, 7.64191687168079254846227617990, 8.848797618304693553014237167959, 10.48634038454108314987431667265, 10.82685091599925790252496530311, 12.00615102256227248919309436921, 13.38531909959618911973881962800

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