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Properties

Label 2-126-7.2-c7-0-7
Degree $2$
Conductor $126$
Sign $0.917 - 0.398i$
Analytic cond. $39.3605$
Root an. cond. $6.27379$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (−219. + 379. i)5-s + (−893. − 159. i)7-s + 511.·8-s + (−1.75e3 − 3.03e3i)10-s + (−2.74e3 − 4.74e3i)11-s + 4.00e3·13-s + (4.68e3 − 5.54e3i)14-s + (−2.04e3 + 3.54e3i)16-s + (−1.40e4 − 2.42e4i)17-s + (−1.19e4 + 2.06e4i)19-s + 2.80e4·20-s + 4.38e4·22-s + (−3.68e4 + 6.38e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.784 + 1.35i)5-s + (−0.984 − 0.176i)7-s + 0.353·8-s + (−0.554 − 0.960i)10-s + (−0.620 − 1.07i)11-s + 0.505·13-s + (0.455 − 0.540i)14-s + (−0.125 + 0.216i)16-s + (−0.691 − 1.19i)17-s + (−0.398 + 0.690i)19-s + 0.784·20-s + 0.877·22-s + (−0.631 + 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(39.3605\)
Root analytic conductor: \(6.27379\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :7/2),\ 0.917 - 0.398i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6819408363\)
\(L(\frac12)\) \(\approx\) \(0.6819408363\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 \)
7 \( 1 + (893. + 159. i)T \)
good5 \( 1 + (219. - 379. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (2.74e3 + 4.74e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 4.00e3T + 6.27e7T^{2} \)
17 \( 1 + (1.40e4 + 2.42e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (1.19e4 - 2.06e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (3.68e4 - 6.38e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 9.87e4T + 1.72e10T^{2} \)
31 \( 1 + (-2.37e4 - 4.11e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-5.00e4 + 8.66e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 4.89e5T + 1.94e11T^{2} \)
43 \( 1 - 2.99e5T + 2.71e11T^{2} \)
47 \( 1 + (-4.81e5 + 8.33e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-9.18e5 - 1.59e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (7.25e3 + 1.25e4i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.01e6 - 1.75e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.48e6 - 2.57e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 4.34e6T + 9.09e12T^{2} \)
73 \( 1 + (7.50e5 + 1.29e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-8.86e5 + 1.53e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 1.57e6T + 2.71e13T^{2} \)
89 \( 1 + (-4.39e6 + 7.61e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.03e7T + 8.07e13T^{2} \)
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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

−11.86801669188938056877275990744, −10.85731519667286810473109165692, −10.10438591633250878235464700377, −8.750317872947281017026755725205, −7.59488898765965087377625331655, −6.75709521933072660312673814392, −5.78987328109233567441657884884, −3.83848895228160037796387975676, −2.81487259951558639872125758452, −0.38942249018279724809480915530, 0.63038654752395917468142909362, 2.21945789622096764319913015185, 3.89170621444564827960455128885, 4.80416127382966785317087302823, 6.53604804164692489720078473348, 8.073396063702544279082094821950, 8.766950776535144281447511502492, 9.822382984234677756452030739479, 10.87896099483356433953265303461, 12.28030391916796443476048888753

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