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Properties

Label 2-126-7.2-c11-0-21
Degree $2$
Conductor $126$
Sign $-0.564 + 0.825i$
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  + (16 − 27.7i)2-s + (−511. − 886. i)4-s + (−2.02e3 + 3.50e3i)5-s + (−4.12e4 + 1.66e4i)7-s − 3.27e4·8-s + (6.48e4 + 1.12e5i)10-s + (−5.26e4 − 9.11e4i)11-s + 1.44e5·13-s + (−1.98e5 + 1.40e6i)14-s + (−5.24e5 + 9.08e5i)16-s + (3.54e6 + 6.14e6i)17-s + (2.86e6 − 4.96e6i)19-s + 4.14e6·20-s − 3.36e6·22-s + (−1.91e7 + 3.31e7i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.289 + 0.502i)5-s + (−0.927 + 0.374i)7-s − 0.353·8-s + (0.205 + 0.355i)10-s + (−0.0985 − 0.170i)11-s + 0.108·13-s + (−0.0985 + 0.700i)14-s + (−0.125 + 0.216i)16-s + (0.605 + 1.04i)17-s + (0.265 − 0.460i)19-s + 0.289·20-s − 0.139·22-s + (−0.619 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(96.8112\)
Root analytic conductor: \(9.83927\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :11/2),\ -0.564 + 0.825i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.016701705\)
\(L(\frac12)\) \(\approx\) \(1.016701705\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16 + 27.7i)T \)
3 \( 1 \)
7 \( 1 + (4.12e4 - 1.66e4i)T \)
good5 \( 1 + (2.02e3 - 3.50e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (5.26e4 + 9.11e4i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 1.44e5T + 1.79e12T^{2} \)
17 \( 1 + (-3.54e6 - 6.14e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (-2.86e6 + 4.96e6i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (1.91e7 - 3.31e7i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 1.21e7T + 1.22e16T^{2} \)
31 \( 1 + (5.57e7 + 9.65e7i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.38e8 + 2.39e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 1.41e9T + 5.50e17T^{2} \)
43 \( 1 - 1.35e9T + 9.29e17T^{2} \)
47 \( 1 + (-3.25e7 + 5.63e7i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (2.08e9 + 3.60e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-4.30e8 - 7.45e8i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (2.59e8 - 4.49e8i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (7.81e9 + 1.35e10i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 - 6.55e9T + 2.31e20T^{2} \)
73 \( 1 + (4.05e9 + 7.02e9i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-1.72e10 + 2.98e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 - 1.62e10T + 1.28e21T^{2} \)
89 \( 1 + (-1.95e10 + 3.38e10i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 + 1.29e10T + 7.15e21T^{2} \)
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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

−10.96761117026176391172663895651, −9.995638629433101525406692864625, −9.059827450658158368234302707211, −7.66435942559567445068052644812, −6.37380143555937262539534173257, −5.41439726762311684534583772671, −3.78733381866688219883538309580, −3.10362598617126381307187106159, −1.77206345431418180397143153171, −0.24446416174055826914943448962, 0.881325910465644355221311248214, 2.80924220213366517693794338947, 3.96453694381847746434424638577, 5.04560663148275439325087674369, 6.26221160811893754447989840616, 7.25281945055601137552641503389, 8.305716382641716063113648497558, 9.425115272759943030585269350094, 10.42960902593103825430756066003, 11.99426949838306992885076102375

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