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Properties

Label 2-126-21.2-c2-0-3
Degree $2$
Conductor $126$
Sign $0.680 - 0.732i$
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 + (0.999 + 1.73i)4-s + (1.22 + 0.707i)5-s + (6.5 + 2.59i)7-s + 2.82i·8-s + (0.999 + 1.73i)10-s + (−6.12 + 3.53i)11-s + 15·13-s + (6.12 + 7.77i)14-s + (−2.00 + 3.46i)16-s + (−9.79 + 5.65i)17-s + (6.5 − 11.2i)19-s + 2.82i·20-s − 10·22-s + (−19.5 − 11.3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.244 + 0.141i)5-s + (0.928 + 0.371i)7-s + 0.353i·8-s + (0.0999 + 0.173i)10-s + (−0.556 + 0.321i)11-s + 1.15·13-s + (0.437 + 0.555i)14-s + (−0.125 + 0.216i)16-s + (−0.576 + 0.332i)17-s + (0.342 − 0.592i)19-s + 0.141i·20-s − 0.454·22-s + (−0.851 − 0.491i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.680 - 0.732i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.680 - 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.92349 + 0.838532i\)
\(L(\frac12)\) \(\approx\) \(1.92349 + 0.838532i\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (-6.5 - 2.59i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.12 - 3.53i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 15T + 169T^{2} \)
17 \( 1 + (9.79 - 5.65i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.5 + 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (19.5 + 11.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 22.6iT - 841T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 80.6iT - 1.68e3T^{2} \)
43 \( 1 + 85T + 1.84e3T^{2} \)
47 \( 1 + (-62.4 - 36.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-29.3 + 16.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (78.3 - 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36 + 62.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.5 + 37.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 52.3iT - 5.04e3T^{2} \)
73 \( 1 + (-47.5 - 82.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (34.5 - 59.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 60.8iT - 6.88e3T^{2} \)
89 \( 1 + (-117. - 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 16T + 9.40e3T^{2} \)
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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

−13.51571813509611265967200204320, −12.31710377010915797822499644537, −11.35278213535930766442512798451, −10.36035234572554778882707046280, −8.756887335646631228562842770883, −7.87632734572414259186465954464, −6.47383219470546976527585392406, −5.37960733032248306767664161930, −4.11551439767430689649463932261, −2.23655606928135104505571122569, 1.61804728909872112504619206933, 3.54997235411080694041214942821, 4.94199475222599745543962838668, 6.03456938735435264446836970504, 7.57403421057571864717543549380, 8.739241601382243938514227346113, 10.16775131510014615197581594796, 11.10041221132107517398053506822, 11.86877353591730060266604036134, 13.30735890555823282895870894939

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