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Properties

Label 2-126-9.5-c2-0-8
Degree $2$
Conductor $126$
Sign $0.585 + 0.810i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.880 − 2.86i)3-s + (0.999 − 1.73i)4-s + (7.45 + 4.30i)5-s + (−0.949 − 4.13i)6-s + (1.32 + 2.29i)7-s − 2.82i·8-s + (−7.44 − 5.05i)9-s + 12.1·10-s + (−12.8 + 7.42i)11-s + (−4.08 − 4.39i)12-s + (2.56 − 4.44i)13-s + (3.24 + 1.87i)14-s + (18.9 − 17.5i)15-s + (−2.00 − 3.46i)16-s − 11.0i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.293 − 0.955i)3-s + (0.249 − 0.433i)4-s + (1.49 + 0.861i)5-s + (−0.158 − 0.689i)6-s + (0.188 + 0.327i)7-s − 0.353i·8-s + (−0.827 − 0.561i)9-s + 1.21·10-s + (−1.16 + 0.674i)11-s + (−0.340 − 0.366i)12-s + (0.197 − 0.341i)13-s + (0.231 + 0.133i)14-s + (1.26 − 1.17i)15-s + (−0.125 − 0.216i)16-s − 0.648i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.12486 - 1.08602i\)
\(L(\frac12)\) \(\approx\) \(2.12486 - 1.08602i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.880 + 2.86i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good5 \( 1 + (-7.45 - 4.30i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (12.8 - 7.42i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.56 + 4.44i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 11.0iT - 289T^{2} \)
19 \( 1 + 31.6T + 361T^{2} \)
23 \( 1 + (-7.82 - 4.51i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-33.5 + 19.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (29.2 - 50.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 16.0T + 1.36e3T^{2} \)
41 \( 1 + (31.0 + 17.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-7.37 - 12.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-70.0 + 40.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 4.52iT - 2.80e3T^{2} \)
59 \( 1 + (-1.90 - 1.09i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.7 - 30.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 + 51.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 40.0iT - 5.04e3T^{2} \)
73 \( 1 + 62.6T + 5.32e3T^{2} \)
79 \( 1 + (-15.2 - 26.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-27.8 + 16.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 61.8iT - 7.92e3T^{2} \)
97 \( 1 + (49.6 + 85.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.10153230222576982545525358668, −12.31935209821342187725717022194, −10.87258482896608930631147144806, −10.12907391537517630436980151168, −8.765792240707072421102980166013, −7.21759735702013843936262899686, −6.26200201379950842003106856824, −5.23900425262389691631767436354, −2.81731202799172212627409833073, −2.02218115163414281952200883109, 2.39984481056699181758763552615, 4.25739622059735602192364322776, 5.31547165622258315839907187390, 6.18072115019109398990149796260, 8.206119171332630028939938303491, 8.980384050311954968029451937733, 10.23352241430454772465385590073, 10.96652497191964341211246022587, 12.76956129789448209835693876367, 13.38565211629209015677966854828

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