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Properties

Label 2-126-63.23-c2-0-15
Degree $2$
Conductor $126$
Sign $-0.889 - 0.456i$
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 + (1.16 − 2.76i)3-s − 2.00·4-s + (−7.41 + 4.28i)5-s + (−3.90 − 1.64i)6-s + (−6.97 + 0.609i)7-s + 2.82i·8-s + (−6.27 − 6.44i)9-s + (6.05 + 10.4i)10-s + (−6.10 − 3.52i)11-s + (−2.33 + 5.52i)12-s + (7.43 − 12.8i)13-s + (0.862 + 9.86i)14-s + (3.18 + 25.4i)15-s + 4.00·16-s + (12.4 − 7.18i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.388 − 0.921i)3-s − 0.500·4-s + (−1.48 + 0.856i)5-s + (−0.651 − 0.274i)6-s + (−0.996 + 0.0871i)7-s + 0.353i·8-s + (−0.697 − 0.716i)9-s + (0.605 + 1.04i)10-s + (−0.555 − 0.320i)11-s + (−0.194 + 0.460i)12-s + (0.571 − 0.990i)13-s + (0.0615 + 0.704i)14-s + (0.212 + 1.69i)15-s + 0.250·16-s + (0.732 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.889 - 0.456i$
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: \((2,\ 126,\ (\ :1),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.107815 + 0.446804i\)
\(L(\frac12)\) \(\approx\) \(0.107815 + 0.446804i\)
\(L(2)\) 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 \( 1 + (-1.16 + 2.76i)T \)
7 \( 1 + (6.97 - 0.609i)T \)
good5 \( 1 + (7.41 - 4.28i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.10 + 3.52i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.43 + 12.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-12.4 + 7.18i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-9.66 + 16.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (34.0 - 19.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (11.7 - 6.80i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 24.1T + 961T^{2} \)
37 \( 1 + (-17.6 + 30.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-7.79 - 4.50i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.4 - 56.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 33.3iT - 2.20e3T^{2} \)
53 \( 1 + (52.4 - 30.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 72.3iT - 3.48e3T^{2} \)
61 \( 1 + 7.98T + 3.72e3T^{2} \)
67 \( 1 + 83.6T + 4.48e3T^{2} \)
71 \( 1 + 61.0iT - 5.04e3T^{2} \)
73 \( 1 + (9.83 + 17.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 10.1T + 6.24e3T^{2} \)
83 \( 1 + (-15.8 + 9.15i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-40.0 - 23.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (49.1 + 85.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.49533417353150447125414092162, −11.61443216101329723028686083405, −10.76138160481464293868464925003, −9.402056244602551148950177296397, −7.991942725493217772252440106007, −7.40446234154425018064550348989, −5.92081187892473819235821183235, −3.56712644400032756516180191562, −2.93920358906313300137690074186, −0.29790762281923147334261633199, 3.65220776744272501264004149972, 4.35439696280179632038498941790, 5.82932489180582909005089208187, 7.54869502374068320569314461222, 8.355471126167296807928307130131, 9.323164178553438619000780765937, 10.37065162184693094172708120571, 11.82513003859300177260991560722, 12.71435813310059516085299850399, 13.93879739827410230349341181936

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