Properties

Label 2-126-7.5-c2-0-5
Degree $2$
Conductor $126$
Sign $-0.989 + 0.144i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−7.24 − 4.18i)5-s + (−6.74 + 1.88i)7-s − 2.82·8-s + (−10.2 + 5.91i)10-s + (3 + 5.19i)11-s − 17.8i·13-s + (−2.46 + 9.58i)14-s + (−2.00 + 3.46i)16-s + (16.2 − 9.37i)17-s + (−14.7 − 8.51i)19-s + 16.7i·20-s + 8.48·22-s + (6.72 − 11.6i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.44 − 0.836i)5-s + (−0.963 + 0.268i)7-s − 0.353·8-s + (−1.02 + 0.591i)10-s + (0.272 + 0.472i)11-s − 1.37i·13-s + (−0.176 + 0.684i)14-s + (−0.125 + 0.216i)16-s + (0.955 − 0.551i)17-s + (−0.775 − 0.447i)19-s + 0.836i·20-s + 0.385·22-s + (0.292 − 0.506i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.989 + 0.144i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.989 + 0.144i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0537770 - 0.739153i\)
\(L(\frac12)\) \(\approx\) \(0.0537770 - 0.739153i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (6.74 - 1.88i)T \)
good5 \( 1 + (7.24 + 4.18i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 17.8iT - 169T^{2} \)
17 \( 1 + (-16.2 + 9.37i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (14.7 + 8.51i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.72 + 11.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (-12.7 + 7.37i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (2.98 - 5.17i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 35.2iT - 1.68e3T^{2} \)
43 \( 1 - 15.4T + 1.84e3T^{2} \)
47 \( 1 + (-28.7 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (17.2 + 29.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (23.6 - 13.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.9 + 20.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-57.1 - 99.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 18.6T + 5.04e3T^{2} \)
73 \( 1 + (101. - 58.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-44.1 + 76.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 75.7iT - 6.88e3T^{2} \)
89 \( 1 + (-18 - 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 30.5iT - 9.40e3T^{2} \)
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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

−12.54541276688741425945192202661, −11.93053035131645276652041567051, −10.73672234666713314625604798567, −9.544254091697781155379272964989, −8.462407142832589518286540870139, −7.27856737918960901944340489852, −5.56361385139454075428257393573, −4.26517285066874729248914854187, −3.10300510715676986415934547553, −0.45728706469460177563151992409, 3.37087450268952251697580910419, 4.13748045058694122429008457519, 6.15613721402608089145109930584, 7.03938743408257599033833084683, 7.967205827201395179097672270862, 9.268959572161725507693014105160, 10.69849217526231397657747095311, 11.71069470592217364712138203689, 12.56539545939959439158337831855, 13.83634994338412684019641177639

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