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Properties

Label 2-126-63.38-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.224 - 0.974i$
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 + (−1.56 + 0.752i)3-s − 4-s + (−1.80 + 3.13i)5-s + (0.752 + 1.56i)6-s + (−2.41 + 1.08i)7-s + i·8-s + (1.86 − 2.34i)9-s + (3.13 + 1.80i)10-s + (−1.73 + 1.00i)11-s + (1.56 − 0.752i)12-s + (2.95 − 1.70i)13-s + (1.08 + 2.41i)14-s + (0.465 − 6.25i)15-s + 16-s + (−3.08 + 5.34i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.900 + 0.434i)3-s − 0.5·4-s + (−0.809 + 1.40i)5-s + (0.307 + 0.636i)6-s + (−0.912 + 0.410i)7-s + 0.353i·8-s + (0.622 − 0.782i)9-s + (0.991 + 0.572i)10-s + (−0.523 + 0.302i)11-s + (0.450 − 0.217i)12-s + (0.818 − 0.472i)13-s + (0.289 + 0.644i)14-s + (0.120 − 1.61i)15-s + 0.250·16-s + (−0.748 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.224 - 0.974i$
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: \((2,\ 126,\ (\ :1/2),\ -0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266158 + 0.334377i\)
\(L(\frac12)\) \(\approx\) \(0.266158 + 0.334377i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 0.752i)T \)
7 \( 1 + (2.41 - 1.08i)T \)
good5 \( 1 + (1.80 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.73 - 1.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.95 + 1.70i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.877 + 0.506i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.62 + 1.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.04 - 2.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.909iT - 31T^{2} \)
37 \( 1 + (-3.66 - 6.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.85 + 4.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.39 - 4.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 + (-7.58 - 4.37i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.98T + 59T^{2} \)
61 \( 1 - 14.7iT - 61T^{2} \)
67 \( 1 + 8.31T + 67T^{2} \)
71 \( 1 + 0.466iT - 71T^{2} \)
73 \( 1 + (3.65 + 2.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
83 \( 1 + (4.00 - 6.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.39 + 4.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.1 + 5.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.35645054724975772113944996865, −12.37453336855485489142821049704, −11.48726225872964170958951254731, −10.54340775737230017577171522137, −10.14661705964147286751606168359, −8.508222430007279177100422427460, −6.89558155748048183595824445016, −5.93606464857169635781630400042, −4.12847877194644150976145584215, −3.02502930330182566012671624345, 0.50425983711148362233818633547, 4.13291167207155487113793990506, 5.22273104064273140634567820061, 6.44937420543007678507631312792, 7.55811067710496087693788998359, 8.610694999673997039525759617096, 9.792934231787109858415276937949, 11.29763452823685716706945294472, 12.18434617640672411911157843855, 13.20026498904350329446812372432

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