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Properties

Label 2-126-63.38-c1-0-6
Degree $2$
Conductor $126$
Sign $0.449 + 0.893i$
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  − i·2-s + (1.64 − 0.533i)3-s − 4-s + (−0.450 + 0.779i)5-s + (−0.533 − 1.64i)6-s + (1.57 − 2.12i)7-s + i·8-s + (2.43 − 1.75i)9-s + (0.779 + 0.450i)10-s + (−2.70 + 1.56i)11-s + (−1.64 + 0.533i)12-s + (−1.99 + 1.14i)13-s + (−2.12 − 1.57i)14-s + (−0.325 + 1.52i)15-s + 16-s + (−2.57 + 4.46i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.951 − 0.308i)3-s − 0.5·4-s + (−0.201 + 0.348i)5-s + (−0.217 − 0.672i)6-s + (0.593 − 0.804i)7-s + 0.353i·8-s + (0.810 − 0.586i)9-s + (0.246 + 0.142i)10-s + (−0.816 + 0.471i)11-s + (−0.475 + 0.154i)12-s + (−0.552 + 0.318i)13-s + (−0.568 − 0.420i)14-s + (−0.0840 + 0.393i)15-s + 0.250·16-s + (−0.624 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.449 + 0.893i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11415 - 0.686444i\)
\(L(\frac12)\) \(\approx\) \(1.11415 - 0.686444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.64 + 0.533i)T \)
7 \( 1 + (-1.57 + 2.12i)T \)
good5 \( 1 + (0.450 - 0.779i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.99 - 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.57 - 4.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 + 1.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.48 + 0.857i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.85 + 1.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + (4.73 + 8.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.22 - 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.273 - 0.473i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.86T + 47T^{2} \)
53 \( 1 + (12.0 + 6.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 + 7.25iT - 61T^{2} \)
67 \( 1 - 3.67T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 + 6.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 + (-0.184 + 0.319i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.00 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.86 - 5.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.16149353005405219912594217735, −12.35489484279368404017787507552, −10.98411614177215929376586710348, −10.21647114696347245679472094370, −9.004944602351424231069393926778, −7.86822885438065826247459862325, −7.00667725130866562794274200155, −4.77366667168282364877367659375, −3.49385296780755240212352942485, −1.94677452765405017883762126328, 2.68844761745597891162081279375, 4.52116217382300115211317411213, 5.54695551408363840865822211141, 7.40286327378922940870618708646, 8.221364898177568409183551196766, 9.051331993824119954071691058403, 10.11026710516933598954924596078, 11.59271105060278165226796764195, 12.83793662791076342837308857368, 13.78225168604499024005751025282

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