↑

Properties

Label 2-126-63.2-c2-0-0
Degree $2$
Conductor $126$
Sign $-0.350 - 0.936i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (−2.66 − 1.37i)3-s + (0.999 − 1.73i)4-s − 2.72i·5-s + (4.23 − 0.207i)6-s + (−4.37 + 5.46i)7-s + 2.82i·8-s + (5.24 + 7.31i)9-s + (1.92 + 3.34i)10-s − 2.91i·11-s + (−5.04 + 3.25i)12-s + (10.4 + 18.1i)13-s + (1.49 − 9.78i)14-s + (−3.73 + 7.27i)15-s + (−2.00 − 3.46i)16-s + (−24.7 + 14.2i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.889 − 0.457i)3-s + (0.249 − 0.433i)4-s − 0.545i·5-s + (0.706 − 0.0346i)6-s + (−0.625 + 0.780i)7-s + 0.353i·8-s + (0.582 + 0.813i)9-s + (0.192 + 0.334i)10-s − 0.264i·11-s + (−0.420 + 0.270i)12-s + (0.805 + 1.39i)13-s + (0.106 − 0.698i)14-s + (−0.249 + 0.485i)15-s + (−0.125 − 0.216i)16-s + (−1.45 + 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.350 - 0.936i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.255685 + 0.368864i\)
\(L(\frac12)\) \(\approx\) \(0.255685 + 0.368864i\)
\(L(2)\) 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 \( 1 + (2.66 + 1.37i)T \)
7 \( 1 + (4.37 - 5.46i)T \)
good5 \( 1 + 2.72iT - 25T^{2} \)
11 \( 1 + 2.91iT - 121T^{2} \)
13 \( 1 + (-10.4 - 18.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (24.7 - 14.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (17.7 - 30.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + 15.7iT - 529T^{2} \)
29 \( 1 + (-18.1 - 10.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.23 + 10.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (5.80 - 10.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-26.4 + 15.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (12.6 - 21.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (73.2 - 42.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (15.1 - 8.77i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (37.3 + 21.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.1 + 26.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (43.2 - 74.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 1.24iT - 5.04e3T^{2} \)
73 \( 1 + (6.48 + 11.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (51.7 + 89.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-35.2 - 20.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (15.5 + 8.95i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (2.62 - 4.54i)T + (-4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.16095759505996875924223154953, −12.41327234136455438376213761769, −11.36021554330148016337625082590, −10.39773729353170624217941633582, −9.015334669632577462091068983262, −8.284243616743553992314375296061, −6.47907150439339588496337902904, −6.18332500186691394500011137479, −4.50368887409262620278906510233, −1.76180490719361802161953471175, 0.41781446947808200925621359262, 3.11357484311845391878798702782, 4.62758213349960304573528136992, 6.40414092329974976015936988398, 7.14930280461051522701248491702, 8.831854643488442595383620072135, 9.974157058281824167638447140290, 10.80036882524349645733547713326, 11.29369109880600017997612665532, 12.77097819408558649737459594927

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