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Properties

Label 2-126-21.17-c11-0-3
Degree $2$
Conductor $126$
Sign $-0.589 - 0.807i$
Analytic cond. $96.8112$
Root an. cond. $9.83927$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (27.7 − 16i)2-s + (511. − 886. i)4-s + (5.24e3 + 9.08e3i)5-s + (4.95e3 − 4.41e4i)7-s − 3.27e4i·8-s + (2.90e5 + 1.67e5i)10-s + (−5.48e5 − 3.16e5i)11-s + 6.95e5i·13-s + (−5.69e5 − 1.30e6i)14-s + (−5.24e5 − 9.08e5i)16-s + (−2.71e6 + 4.69e6i)17-s + (−1.19e7 + 6.88e6i)19-s + 1.07e7·20-s − 2.02e7·22-s + (4.08e7 − 2.35e7i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.750 + 1.29i)5-s + (0.111 − 0.993i)7-s − 0.353i·8-s + (0.918 + 0.530i)10-s + (−1.02 − 0.593i)11-s + 0.519i·13-s + (−0.283 − 0.647i)14-s + (−0.125 − 0.216i)16-s + (−0.463 + 0.802i)17-s + (−1.10 + 0.637i)19-s + 0.750·20-s − 0.838·22-s + (1.32 − 0.764i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.589 - 0.807i$
Analytic conductor: \(96.8112\)
Root analytic conductor: \(9.83927\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :11/2),\ -0.589 - 0.807i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.217657253\)
\(L(\frac12)\) \(\approx\) \(1.217657253\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-27.7 + 16i)T \)
3 \( 1 \)
7 \( 1 + (-4.95e3 + 4.41e4i)T \)
good5 \( 1 + (-5.24e3 - 9.08e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (5.48e5 + 3.16e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 6.95e5iT - 1.79e12T^{2} \)
17 \( 1 + (2.71e6 - 4.69e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.19e7 - 6.88e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-4.08e7 + 2.35e7i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 1.26e8iT - 1.22e16T^{2} \)
31 \( 1 + (2.77e7 + 1.60e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-2.54e8 - 4.40e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 5.07e8T + 5.50e17T^{2} \)
43 \( 1 + 1.35e9T + 9.29e17T^{2} \)
47 \( 1 + (8.56e8 + 1.48e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.98e9 + 1.14e9i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-6.32e8 + 1.09e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (2.07e9 - 1.19e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (7.03e9 - 1.21e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 2.81e10iT - 2.31e20T^{2} \)
73 \( 1 + (-4.02e8 - 2.32e8i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-1.19e9 - 2.06e9i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 3.50e10T + 1.28e21T^{2} \)
89 \( 1 + (2.20e10 + 3.81e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 8.54e10iT - 7.15e21T^{2} \)
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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

−11.29602966105419921430743518569, −10.58976128137242413612160179003, −10.14338539391302641154708792418, −8.421736724211728851311629170473, −6.93174431530118794736099417964, −6.33863687738361288208104094291, −4.96238881586516811991505465194, −3.62939561793512257937527860265, −2.64504418091715909011659045398, −1.52231166931540943749784280819, 0.18453450579451144630718124130, 1.82910699378255724202775104550, 2.79335400725781042094289799283, 4.77037217563864887023839674891, 5.15352648904978827006426732366, 6.23851087717712681962589964444, 7.72869789780214555858300908965, 8.798620126083311877690690693054, 9.570734937310262589918228858851, 11.06952022117041715324721668424

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