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Properties

Label 2-126-63.23-c2-0-1
Degree $2$
Conductor $126$
Sign $-0.981 - 0.193i$
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 + (−0.677 − 2.92i)3-s − 2.00·4-s + (−5.54 + 3.20i)5-s + (4.13 − 0.958i)6-s + (4.41 + 5.43i)7-s − 2.82i·8-s + (−8.08 + 3.96i)9-s + (−4.52 − 7.84i)10-s + (−16.5 − 9.53i)11-s + (1.35 + 5.84i)12-s + (−11.8 + 20.4i)13-s + (−7.68 + 6.24i)14-s + (13.1 + 14.0i)15-s + 4.00·16-s + (−4.77 + 2.75i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.225 − 0.974i)3-s − 0.500·4-s + (−1.10 + 0.640i)5-s + (0.688 − 0.159i)6-s + (0.630 + 0.775i)7-s − 0.353i·8-s + (−0.897 + 0.440i)9-s + (−0.452 − 0.784i)10-s + (−1.50 − 0.867i)11-s + (0.112 + 0.487i)12-s + (−0.909 + 1.57i)13-s + (−0.548 + 0.446i)14-s + (0.874 + 0.935i)15-s + 0.250·16-s + (−0.280 + 0.162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.981 - 0.193i$
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: \((2,\ 126,\ (\ :1),\ -0.981 - 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0341511 + 0.349894i\)
\(L(\frac12)\) \(\approx\) \(0.0341511 + 0.349894i\)
\(L(2)\) 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 \( 1 + (0.677 + 2.92i)T \)
7 \( 1 + (-4.41 - 5.43i)T \)
good5 \( 1 + (5.54 - 3.20i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (16.5 + 9.53i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (11.8 - 20.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (4.77 - 2.75i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.39 + 4.14i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.64 + 3.83i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.7 + 6.79i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 + (-31.7 + 54.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (32.6 + 18.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (3.41 + 5.91i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 83.5iT - 2.20e3T^{2} \)
53 \( 1 + (25.8 - 14.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 28.9iT - 3.48e3T^{2} \)
61 \( 1 + 86.6T + 3.72e3T^{2} \)
67 \( 1 + 64.1T + 4.48e3T^{2} \)
71 \( 1 - 53.5iT - 5.04e3T^{2} \)
73 \( 1 + (-10.1 - 17.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 34.4T + 6.24e3T^{2} \)
83 \( 1 + (12.6 - 7.29i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-71.1 - 41.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (3.32 + 5.75i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.78339561234833718395434599104, −12.55331093849862351134631546139, −11.63006585771840877709120842464, −10.89422984302615739259953987543, −8.927662515293641344153225426711, −7.916592088828505004972683374338, −7.28801599559583448150356705551, −6.04636182657229730364545276960, −4.74025056836412932444272370620, −2.62790435906967618886673913925, 0.24013100927261132309343328992, 3.10982069842431837649107519305, 4.63511581955423404830859346087, 5.04366190419506630237465792486, 7.67254402804001737739112064532, 8.322858255163587023320181715387, 9.984288919958079983443687706901, 10.46723558488123762403136341524, 11.56202372245911353529566866724, 12.42648390587229497123258281621

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