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Properties

Label 2-126-63.32-c2-0-11
Degree 22
Conductor 126126
Sign 0.714+0.699i0.714 + 0.699i
Analytic cond. 3.433253.43325
Root an. cond. 1.852901.85290
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (−2.97 + 0.371i)3-s + (0.999 + 1.73i)4-s − 8.56i·5-s + (−3.90 − 1.64i)6-s + (4.01 − 5.73i)7-s + 2.82i·8-s + (8.72 − 2.21i)9-s + (6.05 − 10.4i)10-s + 7.05i·11-s + (−3.62 − 4.78i)12-s + (7.43 − 12.8i)13-s + (8.97 − 4.18i)14-s + (3.18 + 25.4i)15-s + (−2.00 + 3.46i)16-s + (−12.4 − 7.18i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.992 + 0.123i)3-s + (0.249 + 0.433i)4-s − 1.71i·5-s + (−0.651 − 0.274i)6-s + (0.573 − 0.819i)7-s + 0.353i·8-s + (0.969 − 0.245i)9-s + (0.605 − 1.04i)10-s + 0.641i·11-s + (−0.301 − 0.398i)12-s + (0.571 − 0.990i)13-s + (0.640 − 0.298i)14-s + (0.212 + 1.69i)15-s + (−0.125 + 0.216i)16-s + (−0.732 − 0.422i)17-s + ⋯

Functional equation

Λ(s)=(126s/2 ΓC(s) L(s)=((0.714+0.699i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(126s/2 ΓC(s+1) L(s)=((0.714+0.699i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 126126    =    2⋅32⋅72 \cdot 3^{2} \cdot 7
Sign: 0.714+0.699i0.714 + 0.699i
Analytic conductor: 3.433253.43325
Root analytic conductor: 1.852901.85290
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ126(95,⋅)\chi_{126} (95, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 126, ( :1), 0.714+0.699i)(2,\ 126,\ (\ :1),\ 0.714 + 0.699i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.36888−0.558493i1.36888 - 0.558493i
L(12)L(\frac12) ≈\approx 1.36888−0.558493i1.36888 - 0.558493i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−1.22−0.707i)T 1 + (-1.22 - 0.707i)T
3 1+(2.97−0.371i)T 1 + (2.97 - 0.371i)T
7 1+(−4.01+5.73i)T 1 + (-4.01 + 5.73i)T
good5 1+8.56iT−25T2 1 + 8.56iT - 25T^{2}
11 1−7.05iT−121T2 1 - 7.05iT - 121T^{2}
13 1+(−7.43+12.8i)T+(−84.5−146.i)T2 1 + (-7.43 + 12.8i)T + (-84.5 - 146. i)T^{2}
17 1+(12.4+7.18i)T+(144.5+250.i)T2 1 + (12.4 + 7.18i)T + (144.5 + 250. i)T^{2}
19 1+(−9.66−16.7i)T+(−180.5+312.i)T2 1 + (-9.66 - 16.7i)T + (-180.5 + 312. i)T^{2}
23 1+39.3iT−529T2 1 + 39.3iT - 529T^{2}
29 1+(11.7−6.80i)T+(420.5−728.i)T2 1 + (11.7 - 6.80i)T + (420.5 - 728. i)T^{2}
31 1+(−12.0−20.8i)T+(−480.5+832.i)T2 1 + (-12.0 - 20.8i)T + (-480.5 + 832. i)T^{2}
37 1+(−17.6−30.5i)T+(−684.5+1.18e3i)T2 1 + (-17.6 - 30.5i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−7.79−4.50i)T+(840.5+1.45e3i)T2 1 + (-7.79 - 4.50i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−32.4−56.2i)T+(−924.5+1.60e3i)T2 1 + (-32.4 - 56.2i)T + (-924.5 + 1.60e3i)T^{2}
47 1+(−28.9−16.6i)T+(1.10e3+1.91e3i)T2 1 + (-28.9 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2}
53 1+(−52.4−30.2i)T+(1.40e3+2.43e3i)T2 1 + (-52.4 - 30.2i)T + (1.40e3 + 2.43e3i)T^{2}
59 1+(62.6−36.1i)T+(1.74e3−3.01e3i)T2 1 + (62.6 - 36.1i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−3.99+6.91i)T+(−1.86e3−3.22e3i)T2 1 + (-3.99 + 6.91i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−41.8−72.4i)T+(−2.24e3+3.88e3i)T2 1 + (-41.8 - 72.4i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+61.0iT−5.04e3T2 1 + 61.0iT - 5.04e3T^{2}
73 1+(9.83−17.0i)T+(−2.66e3−4.61e3i)T2 1 + (9.83 - 17.0i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−5.09+8.82i)T+(−3.12e3−5.40e3i)T2 1 + (-5.09 + 8.82i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−15.8+9.15i)T+(3.44e3−5.96e3i)T2 1 + (-15.8 + 9.15i)T + (3.44e3 - 5.96e3i)T^{2}
89 1+(40.0−23.1i)T+(3.96e3−6.85e3i)T2 1 + (40.0 - 23.1i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(49.1+85.1i)T+(−4.70e3+8.14e3i)T2 1 + (49.1 + 85.1i)T + (-4.70e3 + 8.14e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.79931684698348193201809758737, −12.32446245054543953597406368564, −11.15439312928044761680322391456, −10.03775671315888042179308187997, −8.562894345754972877089413259801, −7.47998545251227795980294746690, −6.00966672174142014502513408449, −4.86991588381691882154501112322, −4.30236515473478709739661559302, −1.05861427945241083459247731185, 2.18301542618570927895562844812, 3.86367861381910705725530066262, 5.54634462409805606387295749554, 6.38677222243108101045271158029, 7.40732850669595698015930423675, 9.346951332945848526693499131445, 10.77153653273469338678598094478, 11.30415002070760890947208681400, 11.79793715259879522570290088743, 13.35842981852306864682135232397

Graph of the ZZ-function along the critical line