Properties

Label 2-126-63.38-c1-0-3
Degree $2$
Conductor $126$
Sign $0.577 + 0.816i$
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 + (−1.08 + 1.35i)3-s − 4-s + (1.77 − 3.07i)5-s + (1.35 + 1.08i)6-s + (2.63 + 0.201i)7-s + i·8-s + (−0.645 − 2.92i)9-s + (−3.07 − 1.77i)10-s + (2.61 − 1.51i)11-s + (1.08 − 1.35i)12-s + (−0.888 + 0.513i)13-s + (0.201 − 2.63i)14-s + (2.22 + 5.73i)15-s + 16-s + (−0.809 + 1.40i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.626 + 0.779i)3-s − 0.5·4-s + (0.794 − 1.37i)5-s + (0.551 + 0.442i)6-s + (0.997 + 0.0762i)7-s + 0.353i·8-s + (−0.215 − 0.976i)9-s + (−0.972 − 0.561i)10-s + (0.789 − 0.455i)11-s + (0.313 − 0.389i)12-s + (−0.246 + 0.142i)13-s + (0.0539 − 0.705i)14-s + (0.574 + 1.48i)15-s + 0.250·16-s + (−0.196 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.577 + 0.816i$
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: \((2,\ 126,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900845 - 0.465906i\)
\(L(\frac12)\) \(\approx\) \(0.900845 - 0.465906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.35i)T \)
7 \( 1 + (-2.63 - 0.201i)T \)
good5 \( 1 + (-1.77 + 3.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.61 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.888 - 0.513i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.809 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.12 - 4.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.90 - 1.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.70 + 2.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.98iT - 31T^{2} \)
37 \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.14T + 47T^{2} \)
53 \( 1 + (2.76 + 1.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.84T + 59T^{2} \)
61 \( 1 - 4.69iT - 61T^{2} \)
67 \( 1 + 0.375T + 67T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.924T + 79T^{2} \)
83 \( 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.35 + 4.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.3 - 7.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.93294270039203719857961070784, −12.10854043456195814974518512577, −11.18542894472985576656024133694, −10.18378430634614354079767902926, −9.093203231390590879796908047925, −8.481743281550610131894952886533, −6.07118058658196596784788366148, −5.02101313287783859930425530226, −4.11900023849943591704680287651, −1.52723067066618501694481823347, 2.19737931407136871382519441899, 4.70553179126040464194364107822, 6.11826326850018767392095687558, 6.83673457466841323799126016092, 7.73802044906836558123383288527, 9.242578802925028920533407013865, 10.70718687708100796870761623957, 11.27238493348260712114890407503, 12.71202129380532576198141397334, 13.70228106350918483040836110392

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