↑

Properties

Label 2-126-21.17-c7-0-19
Degree $2$
Conductor $126$
Sign $-0.647 - 0.761i$
Analytic cond. $39.3605$
Root an. cond. $6.27379$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.92 − 4i)2-s + (31.9 − 55.4i)4-s + (−214. − 372. i)5-s + (34.0 − 906. i)7-s − 511. i·8-s + (−2.97e3 − 1.71e3i)10-s + (−5.00e3 − 2.88e3i)11-s + 5.95e3i·13-s + (−3.39e3 − 6.41e3i)14-s + (−2.04e3 − 3.54e3i)16-s + (−9.36e3 + 1.62e4i)17-s + (3.64e4 − 2.10e4i)19-s − 2.75e4·20-s − 4.62e4·22-s + (3.87e4 − 2.23e4i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.768 − 1.33i)5-s + (0.0375 − 0.999i)7-s − 0.353i·8-s + (−0.941 − 0.543i)10-s + (−1.13 − 0.654i)11-s + 0.751i·13-s + (−0.330 − 0.625i)14-s + (−0.125 − 0.216i)16-s + (−0.462 + 0.800i)17-s + (1.21 − 0.703i)19-s − 0.768·20-s − 0.925·22-s + (0.663 − 0.383i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 - 0.761i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.647 - 0.761i$
Analytic conductor: \(39.3605\)
Root analytic conductor: \(6.27379\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :7/2),\ -0.647 - 0.761i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.002435790\)
\(L(\frac12)\) \(\approx\) \(1.002435790\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-6.92 + 4i)T \)
3 \( 1 \)
7 \( 1 + (-34.0 + 906. i)T \)
good5 \( 1 + (214. + 372. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (5.00e3 + 2.88e3i)T + (9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 5.95e3iT - 6.27e7T^{2} \)
17 \( 1 + (9.36e3 - 1.62e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-3.64e4 + 2.10e4i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-3.87e4 + 2.23e4i)T + (1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 2.23e5iT - 1.72e10T^{2} \)
31 \( 1 + (6.16e4 + 3.56e4i)T + (1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-2.49e4 - 4.32e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 3.32e5T + 1.94e11T^{2} \)
43 \( 1 - 5.40e5T + 2.71e11T^{2} \)
47 \( 1 + (2.94e5 + 5.10e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (1.42e6 + 8.21e5i)T + (5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-2.28e4 + 3.96e4i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.90e6 - 1.10e6i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (8.23e5 - 1.42e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 5.39e6iT - 9.09e12T^{2} \)
73 \( 1 + (-3.67e6 - 2.11e6i)T + (5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-2.44e6 - 4.24e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 6.95e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.14e6 - 3.72e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.40e7iT - 8.07e13T^{2} \)
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

−11.39750237507642003811179678612, −10.65994515834616660680817570869, −9.201883089835845520114802419087, −8.137562432711579307739927665319, −6.98650138415574330403461824485, −5.26151763714428551905653818361, −4.48095788689125647740619361555, −3.30926070324500045130122955944, −1.34601966071318566029568753626, −0.23538363672229272363119506555, 2.50841290401541866441673753416, 3.27250125438540732649939732298, 4.92169025655611606625685721465, 6.03007618102764446861107485381, 7.37567912543505941778145542832, 7.916590905739425884844423354103, 9.628821047971314238353677271753, 10.87332883343434768958733785363, 11.71223355376114696388511025506, 12.62775122846608142230719218236

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