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Properties

Label 2-405-45.13-c2-0-40
Degree 22
Conductor 405405
Sign −0.176+0.984i-0.176 + 0.984i
Analytic cond. 11.035411.0354
Root an. cond. 3.321963.32196
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  + (2.15 + 0.578i)2-s + (0.866 + 0.5i)4-s + (−3.31 − 3.74i)5-s + (6.83 + 1.83i)7-s + (−4.74 − 4.74i)8-s + (−5 − 10i)10-s + (−7.90 − 13.6i)11-s + (−13.6 + 3.66i)13-s + (13.6 + 7.90i)14-s + (−9.49 − 16.4i)16-s + (3.16 − 3.16i)17-s − 18i·19-s + (−1.00 − 4.89i)20-s + (−9.15 − 34.1i)22-s + (−4.31 + 1.15i)23-s + ⋯
L(s)  = 1  + (1.07 + 0.289i)2-s + (0.216 + 0.125i)4-s + (−0.663 − 0.748i)5-s + (0.975 + 0.261i)7-s + (−0.592 − 0.592i)8-s + (−0.5 − i)10-s + (−0.718 − 1.24i)11-s + (−1.05 + 0.281i)13-s + (0.978 + 0.564i)14-s + (−0.593 − 1.02i)16-s + (0.186 − 0.186i)17-s − 0.947i·19-s + (−0.0501 − 0.244i)20-s + (−0.415 − 1.55i)22-s + (−0.187 + 0.0503i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.176+0.984i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1) L(s)=((−0.176+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.176+0.984i-0.176 + 0.984i
Analytic conductor: 11.035411.0354
Root analytic conductor: 3.321963.32196
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ405(28,⋅)\chi_{405} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1), −0.176+0.984i)(2,\ 405,\ (\ :1),\ -0.176 + 0.984i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.17472−1.40377i1.17472 - 1.40377i
L(12)L(\frac12) ≈\approx 1.17472−1.40377i1.17472 - 1.40377i
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)
bad3 1 1
5 1+(3.31+3.74i)T 1 + (3.31 + 3.74i)T
good2 1+(−2.15−0.578i)T+(3.46+2i)T2 1 + (-2.15 - 0.578i)T + (3.46 + 2i)T^{2}
7 1+(−6.83−1.83i)T+(42.4+24.5i)T2 1 + (-6.83 - 1.83i)T + (42.4 + 24.5i)T^{2}
11 1+(7.90+13.6i)T+(−60.5+104.i)T2 1 + (7.90 + 13.6i)T + (-60.5 + 104. i)T^{2}
13 1+(13.6−3.66i)T+(146.−84.5i)T2 1 + (13.6 - 3.66i)T + (146. - 84.5i)T^{2}
17 1+(−3.16+3.16i)T−289iT2 1 + (-3.16 + 3.16i)T - 289iT^{2}
19 1+18iT−361T2 1 + 18iT - 361T^{2}
23 1+(4.31−1.15i)T+(458.−264.5i)T2 1 + (4.31 - 1.15i)T + (458. - 264.5i)T^{2}
29 1+(−41.0+23.7i)T+(420.5−728.i)T2 1 + (-41.0 + 23.7i)T + (420.5 - 728. i)T^{2}
31 1+(4−6.92i)T+(−480.5−832.i)T2 1 + (4 - 6.92i)T + (-480.5 - 832. i)T^{2}
37 1+(−10+10i)T−1.36e3iT2 1 + (-10 + 10i)T - 1.36e3iT^{2}
41 1+(−15.8+27.3i)T+(−840.5−1.45e3i)T2 1 + (-15.8 + 27.3i)T + (-840.5 - 1.45e3i)T^{2}
43 1+(−3.66+13.6i)T+(−1.60e3−924.5i)T2 1 + (-3.66 + 13.6i)T + (-1.60e3 - 924.5i)T^{2}
47 1+(56.1+15.0i)T+(1.91e3+1.10e3i)T2 1 + (56.1 + 15.0i)T + (1.91e3 + 1.10e3i)T^{2}
53 1+(25.2+25.2i)T+2.80e3iT2 1 + (25.2 + 25.2i)T + 2.80e3iT^{2}
59 1+(−41.0−23.7i)T+(1.74e3+3.01e3i)T2 1 + (-41.0 - 23.7i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−29−50.2i)T+(−1.86e3+3.22e3i)T2 1 + (-29 - 50.2i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−25.6−95.6i)T+(−3.88e3+2.24e3i)T2 1 + (-25.6 - 95.6i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1−63.2T+5.04e3T2 1 - 63.2T + 5.04e3T^{2}
73 1+(−55−55i)T+5.32e3iT2 1 + (-55 - 55i)T + 5.32e3iT^{2}
79 1+(10.3−6i)T+(3.12e3−5.40e3i)T2 1 + (10.3 - 6i)T + (3.12e3 - 5.40e3i)T^{2}
83 1+(19.6−73.4i)T+(−5.96e3−3.44e3i)T2 1 + (19.6 - 73.4i)T + (-5.96e3 - 3.44e3i)T^{2}
89 1−7.92e3T2 1 - 7.92e3T^{2}
97 1+(−6.83−1.83i)T+(8.14e3+4.70e3i)T2 1 + (-6.83 - 1.83i)T + (8.14e3 + 4.70e3i)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

−11.22472552585445105783883174010, −9.834706008646479925558087178881, −8.721963373734073360052043065788, −8.042857356724882133434395128832, −6.89770848509319085774816139785, −5.50170623691002039660780574756, −4.99501916716361697928433680631, −4.10581530603600100404992290225, −2.73108853449954190467969906399, −0.54715903863425491032257209176, 2.17756299101604149336355291148, 3.30051113190840078345788710464, 4.54641298182923400719600449977, 5.00479386010629767371990815981, 6.46370384164709270136836121304, 7.70251528290692197337721884145, 8.117588985210602963336378824402, 9.783415483716555211529616640188, 10.62165611856763363431691020196, 11.49945837824783245066322110258

Graph of the ZZ-function along the critical line