Properties

Label 2-126-63.41-c1-0-1
Degree $2$
Conductor $126$
Sign $0.723 - 0.690i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (1.62 − 0.608i)3-s + (0.499 − 0.866i)4-s + (−1.94 + 3.36i)5-s + (−1.10 + 1.33i)6-s + (2.09 + 1.60i)7-s + 0.999i·8-s + (2.26 − 1.97i)9-s − 3.89i·10-s + (3.41 − 1.97i)11-s + (0.284 − 1.70i)12-s + (−2.46 − 1.42i)13-s + (−2.62 − 0.343i)14-s + (−1.10 + 6.64i)15-s + (−0.5 − 0.866i)16-s − 0.742·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.936 − 0.351i)3-s + (0.249 − 0.433i)4-s + (−0.870 + 1.50i)5-s + (−0.449 + 0.546i)6-s + (0.793 + 0.608i)7-s + 0.353i·8-s + (0.753 − 0.657i)9-s − 1.23i·10-s + (1.03 − 0.594i)11-s + (0.0820 − 0.493i)12-s + (−0.684 − 0.395i)13-s + (−0.701 − 0.0919i)14-s + (−0.285 + 1.71i)15-s + (−0.125 − 0.216i)16-s − 0.179·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.949782 + 0.380215i\)
\(L(\frac12)\) \(\approx\) \(0.949782 + 0.380215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.62 + 0.608i)T \)
7 \( 1 + (-2.09 - 1.60i)T \)
good5 \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.41 + 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.46 + 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.742T + 17T^{2} \)
19 \( 1 - 1.78iT - 19T^{2} \)
23 \( 1 + (5.41 + 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.50 - 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.04 + 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.09 + 1.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.0105 - 0.0183i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.72 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 + (1.81 + 3.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.02 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 + (16.2 - 9.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.08457116060493978017432847451, −12.24540398019516900574044445913, −11.39072086517669874310171684431, −10.32538680004834245497628924998, −9.030311391300753122748796099716, −8.013113512560948855018556327236, −7.31272016212396169951795407072, −6.18043132116941745820290339410, −3.85155398602866355139392476605, −2.37252111643370981445405582503, 1.62019668701832695227740765842, 3.98768681714196133660270242315, 4.66426106015470652096624380482, 7.36927025757613417420469724718, 8.070527265685172916323998125139, 9.066950412843359688014421538420, 9.739593061322474448266270359184, 11.27937732185803586454744386805, 12.12037376030140920778037297357, 13.12756512040172887931615263406

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