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Properties

Label 2-126-63.47-c1-0-5
Degree $2$
Conductor $126$
Sign $0.423 + 0.905i$
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 + (0.361 − 1.69i)3-s + (0.499 − 0.866i)4-s − 0.900·5-s + (0.533 + 1.64i)6-s + (1.05 − 2.42i)7-s + 0.999i·8-s + (−2.73 − 1.22i)9-s + (0.779 − 0.450i)10-s − 3.12i·11-s + (−1.28 − 1.16i)12-s + (1.99 − 1.14i)13-s + (0.296 + 2.62i)14-s + (−0.325 + 1.52i)15-s + (−0.5 − 0.866i)16-s + (2.57 + 4.46i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.208 − 0.977i)3-s + (0.249 − 0.433i)4-s − 0.402·5-s + (0.217 + 0.672i)6-s + (0.399 − 0.916i)7-s + 0.353i·8-s + (−0.912 − 0.408i)9-s + (0.246 − 0.142i)10-s − 0.943i·11-s + (−0.371 − 0.334i)12-s + (0.552 − 0.318i)13-s + (0.0793 + 0.702i)14-s + (−0.0840 + 0.393i)15-s + (−0.125 − 0.216i)16-s + (0.624 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.423 + 0.905i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.423 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694119 - 0.441530i\)
\(L(\frac12)\) \(\approx\) \(0.694119 - 0.441530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.361 + 1.69i)T \)
7 \( 1 + (-1.05 + 2.42i)T \)
good5 \( 1 + 0.900T + 5T^{2} \)
11 \( 1 + 3.12iT - 11T^{2} \)
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 \( 1 - 1.71iT - 23T^{2} \)
29 \( 1 + (1.85 + 1.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.66 - 5.00i)T + (15.5 + 26.8i)T^{2} \)
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 \( 1 + (-3.93 - 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.0 + 6.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.99 + 6.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.28 + 3.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 - 3.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 - 6.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.27 - 5.67i)T + (-39.5 + 68.4i)T^{2} \)
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

−13.45916041417738365976641151453, −12.05573967177817065538576494774, −11.15121383007636456073638138186, −10.07704607370002274371863889582, −8.384364672634128193009128470533, −8.021529114076941072643945492666, −6.84026599826854462116751193619, −5.69463698092370923691070776129, −3.52208762919151882527043877517, −1.19786241863204647696261456582, 2.55809514809773299384590866064, 4.15439464904444938749923270517, 5.51052372565232326992470783228, 7.39974548310578898268151966271, 8.558504780614642967803875598814, 9.393556993913676335417994805780, 10.28942549606219884043566494689, 11.59098746401789048884376689306, 11.97552704068566260719212102476, 13.63814598665946019208790301955

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