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Properties

Label 2-126-63.41-c3-0-18
Degree $2$
Conductor $126$
Sign $-0.993 + 0.109i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − i)2-s + (−4.36 + 2.81i)3-s + (1.99 − 3.46i)4-s + (−5.62 + 9.74i)5-s + (−4.75 + 9.24i)6-s + (2.97 − 18.2i)7-s − 7.99i·8-s + (11.1 − 24.5i)9-s + 22.5i·10-s + (−52.7 + 30.4i)11-s + (1.00 + 20.7i)12-s + (−73.7 − 42.5i)13-s + (−13.1 − 34.6i)14-s + (−2.83 − 58.4i)15-s + (−8 − 13.8i)16-s − 54.9·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.840 + 0.541i)3-s + (0.249 − 0.433i)4-s + (−0.503 + 0.871i)5-s + (−0.323 + 0.628i)6-s + (0.160 − 0.986i)7-s − 0.353i·8-s + (0.413 − 0.910i)9-s + 0.711i·10-s + (−1.44 + 0.834i)11-s + (0.0242 + 0.499i)12-s + (−1.57 − 0.907i)13-s + (−0.250 − 0.661i)14-s + (−0.0488 − 1.00i)15-s + (−0.125 − 0.216i)16-s − 0.784·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.993 + 0.109i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00674108 - 0.122273i\)
\(L(\frac12)\) \(\approx\) \(0.00674108 - 0.122273i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 + (4.36 - 2.81i)T \)
7 \( 1 + (-2.97 + 18.2i)T \)
good5 \( 1 + (5.62 - 9.74i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (52.7 - 30.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (73.7 + 42.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 54.9T + 4.91e3T^{2} \)
19 \( 1 - 9.90iT - 6.85e3T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-204. + 117. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (53.1 + 30.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 + (36.9 - 63.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-116. - 201. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (43.8 + 76.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 56.2iT - 1.48e5T^{2} \)
59 \( 1 + (217. - 376. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-715. + 413. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (451. - 782. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 894. iT - 3.57e5T^{2} \)
73 \( 1 + 629. iT - 3.89e5T^{2} \)
79 \( 1 + (469. + 812. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-236. - 409. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 417.T + 7.04e5T^{2} \)
97 \( 1 + (-485. + 280. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.34079605536381283447241659408, −11.27749844735100468031790291797, −10.36291533336206150736705720492, −10.04884441083267738357651244534, −7.63481837435719538901277864380, −6.85485270457257082648536238017, −5.21841491383388118540301333301, −4.35038053228639250685710448449, −2.81876660644383083121000604931, −0.05322235520188697056976961500, 2.40271018746704751262903357364, 4.79822235719482796422424791324, 5.27824530483502987839969093243, 6.68600390253382271613583729575, 7.88054535082959723691537271869, 8.847126628463052524082761224472, 10.63339440361242570801742902427, 11.81873232031314532235919400858, 12.33407994742207530163978198608, 13.12822180332399154103645844285

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