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Properties

Label 2-126-63.23-c2-0-6
Degree $2$
Conductor $126$
Sign $0.910 - 0.414i$
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  + 1.41i·2-s + (−2.92 − 0.657i)3-s − 2.00·4-s + (2.90 − 1.67i)5-s + (0.929 − 4.13i)6-s + (5.64 − 4.13i)7-s − 2.82i·8-s + (8.13 + 3.84i)9-s + (2.37 + 4.10i)10-s + (12.2 + 7.09i)11-s + (5.85 + 1.31i)12-s + (−8.31 + 14.4i)13-s + (5.84 + 7.99i)14-s + (−9.59 + 2.99i)15-s + 4.00·16-s + (13.5 − 7.82i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.975 − 0.219i)3-s − 0.500·4-s + (0.580 − 0.335i)5-s + (0.154 − 0.689i)6-s + (0.807 − 0.590i)7-s − 0.353i·8-s + (0.903 + 0.427i)9-s + (0.237 + 0.410i)10-s + (1.11 + 0.644i)11-s + (0.487 + 0.109i)12-s + (−0.639 + 1.10i)13-s + (0.417 + 0.570i)14-s + (−0.639 + 0.199i)15-s + 0.250·16-s + (0.797 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19811 + 0.260066i\)
\(L(\frac12)\) \(\approx\) \(1.19811 + 0.260066i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (2.92 + 0.657i)T \)
7 \( 1 + (-5.64 + 4.13i)T \)
good5 \( 1 + (-2.90 + 1.67i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-12.2 - 7.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (8.31 - 14.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-13.5 + 7.82i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-17.3 + 29.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-22.7 + 13.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.2 + 10.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 33.7T + 961T^{2} \)
37 \( 1 + (31.9 - 55.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-5.44 - 3.14i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.0742 - 0.128i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 41.9iT - 2.20e3T^{2} \)
53 \( 1 + (38.6 - 22.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 41.4iT - 3.48e3T^{2} \)
61 \( 1 - 15.9T + 3.72e3T^{2} \)
67 \( 1 + 85.3T + 4.48e3T^{2} \)
71 \( 1 - 56.4iT - 5.04e3T^{2} \)
73 \( 1 + (-4.42 - 7.67i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 29.1T + 6.24e3T^{2} \)
83 \( 1 + (48.8 - 28.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (5.43 + 3.13i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (32.1 + 55.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.38167830284458340660576762047, −12.08931768032343088375989892232, −11.35409973507180120424731209694, −9.925552028898611595686453873721, −9.061119952871595227043302173468, −7.29776089612343811761845319330, −6.79619947117317800844925702477, −5.22162265902588412897104413875, −4.52413618399836602855857488447, −1.31639123463403166692022219998, 1.42525773791148910211764277748, 3.51911283232801811403269870220, 5.25533427591192586008811796610, 5.94913198972285384846111816013, 7.70415375630169221710242505802, 9.203358201243132575253247978831, 10.19758916934055726510004046002, 11.01717197462723406640947188614, 12.01601712367581880924521642159, 12.59530732473019320872247636983

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