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Properties

Label 2-126-21.11-c2-0-1
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: $\chi_{126} (53, \cdot )$
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

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

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