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Properties

Label 2-126-21.11-c2-0-3
Degree $2$
Conductor $126$
Sign $0.524 + 0.851i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (3.67 − 2.12i)5-s + (−3.5 − 6.06i)7-s − 2.82i·8-s + (3 − 5.19i)10-s + (11.0 + 6.36i)11-s − 13-s + (−8.57 − 4.94i)14-s + (−2.00 − 3.46i)16-s + (14.6 + 8.48i)17-s + (−11.5 − 19.9i)19-s − 8.48i·20-s + 18·22-s + (−14.6 + 8.48i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.734 − 0.424i)5-s + (−0.5 − 0.866i)7-s − 0.353i·8-s + (0.300 − 0.519i)10-s + (1.00 + 0.578i)11-s − 0.0769·13-s + (−0.612 − 0.353i)14-s + (−0.125 − 0.216i)16-s + (0.864 + 0.499i)17-s + (−0.605 − 1.04i)19-s − 0.424i·20-s + 0.818·22-s + (−0.638 + 0.368i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81108 - 1.01148i\)
\(L(\frac12)\) \(\approx\) \(1.81108 - 1.01148i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (3.5 + 6.06i)T \)
good5 \( 1 + (-3.67 + 2.12i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-11.0 - 6.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + T + 169T^{2} \)
17 \( 1 + (-14.6 - 8.48i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (11.5 + 19.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (14.6 - 8.48i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 33.9iT - 841T^{2} \)
31 \( 1 + (23.5 - 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-27.5 - 47.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 - 23T + 1.84e3T^{2} \)
47 \( 1 + (3.67 - 2.12i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-44.0 - 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (73.4 + 42.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (52 + 90.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-48.5 + 84.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 97.5iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 - 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (56.5 + 97.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 + (-117. + 67.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 104T + 9.40e3T^{2} \)
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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

−12.94830219858414341989624141353, −12.22992907340983774769136614355, −10.88785252597332281766649603714, −9.932839859397612004920726444335, −9.047645845221223444874556426567, −7.23526450570688817161070108714, −6.19178104287415572690315553211, −4.82183841278427246799468829340, −3.52607685136252793837716336924, −1.52330927143123835174096723099, 2.41770157712237887796399919123, 3.93557655546868593158299665112, 5.88063584730811461727270881512, 6.17068236854293091036916242674, 7.78448208418965535113255957010, 9.132350623213221004722475238012, 10.08676892452193269604250304756, 11.54743804670179957885519682346, 12.34515591173734629527441885350, 13.43234975465333381633831589063

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