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Properties

Label 2-45-45.14-c2-0-6
Degree 22
Conductor 4545
Sign 0.995+0.0980i0.995 + 0.0980i
Analytic cond. 1.226161.22616
Root an. cond. 1.107321.10732
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  + (0.668 + 1.15i)2-s + (−0.320 − 2.98i)3-s + (1.10 − 1.91i)4-s + (4.41 + 2.33i)5-s + (3.24 − 2.36i)6-s + (−7.10 + 4.10i)7-s + 8.30·8-s + (−8.79 + 1.91i)9-s + (0.244 + 6.68i)10-s + (−5.67 + 3.27i)11-s + (−6.06 − 2.68i)12-s + (−1.29 − 0.749i)13-s + (−9.50 − 5.49i)14-s + (5.56 − 13.9i)15-s + (1.13 + 1.97i)16-s + 15.1·17-s + ⋯
L(s)  = 1  + (0.334 + 0.579i)2-s + (−0.106 − 0.994i)3-s + (0.276 − 0.478i)4-s + (0.883 + 0.467i)5-s + (0.540 − 0.394i)6-s + (−1.01 + 0.586i)7-s + 1.03·8-s + (−0.977 + 0.212i)9-s + (0.0244 + 0.668i)10-s + (−0.515 + 0.297i)11-s + (−0.505 − 0.223i)12-s + (−0.0998 − 0.0576i)13-s + (−0.679 − 0.392i)14-s + (0.370 − 0.928i)15-s + (0.0710 + 0.123i)16-s + 0.889·17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.995+0.0980i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1) L(s)=((0.995+0.0980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.995+0.0980i0.995 + 0.0980i
Analytic conductor: 1.226161.22616
Root analytic conductor: 1.107321.10732
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ45(14,⋅)\chi_{45} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1), 0.995+0.0980i)(2,\ 45,\ (\ :1),\ 0.995 + 0.0980i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.35022−0.0663292i1.35022 - 0.0663292i
L(12)L(\frac12) ≈\approx 1.35022−0.0663292i1.35022 - 0.0663292i
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+(0.320+2.98i)T 1 + (0.320 + 2.98i)T
5 1+(−4.41−2.33i)T 1 + (-4.41 - 2.33i)T
good2 1+(−0.668−1.15i)T+(−2+3.46i)T2 1 + (-0.668 - 1.15i)T + (-2 + 3.46i)T^{2}
7 1+(7.10−4.10i)T+(24.5−42.4i)T2 1 + (7.10 - 4.10i)T + (24.5 - 42.4i)T^{2}
11 1+(5.67−3.27i)T+(60.5−104.i)T2 1 + (5.67 - 3.27i)T + (60.5 - 104. i)T^{2}
13 1+(1.29+0.749i)T+(84.5+146.i)T2 1 + (1.29 + 0.749i)T + (84.5 + 146. i)T^{2}
17 1−15.1T+289T2 1 - 15.1T + 289T^{2}
19 1+25.9T+361T2 1 + 25.9T + 361T^{2}
23 1+(11.6−20.1i)T+(−264.5−458.i)T2 1 + (11.6 - 20.1i)T + (-264.5 - 458. i)T^{2}
29 1+(−6.96+4.02i)T+(420.5−728.i)T2 1 + (-6.96 + 4.02i)T + (420.5 - 728. i)T^{2}
31 1+(−22.5+38.9i)T+(−480.5−832.i)T2 1 + (-22.5 + 38.9i)T + (-480.5 - 832. i)T^{2}
37 1+62.8iT−1.36e3T2 1 + 62.8iT - 1.36e3T^{2}
41 1+(−9.97−5.75i)T+(840.5+1.45e3i)T2 1 + (-9.97 - 5.75i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−36.9+21.3i)T+(924.5−1.60e3i)T2 1 + (-36.9 + 21.3i)T + (924.5 - 1.60e3i)T^{2}
47 1+(−8.25−14.2i)T+(−1.10e3+1.91e3i)T2 1 + (-8.25 - 14.2i)T + (-1.10e3 + 1.91e3i)T^{2}
53 1+66.0T+2.80e3T2 1 + 66.0T + 2.80e3T^{2}
59 1+(−0.373−0.215i)T+(1.74e3+3.01e3i)T2 1 + (-0.373 - 0.215i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−15.7−27.3i)T+(−1.86e3+3.22e3i)T2 1 + (-15.7 - 27.3i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(83.1+47.9i)T+(2.24e3+3.88e3i)T2 1 + (83.1 + 47.9i)T + (2.24e3 + 3.88e3i)T^{2}
71 1+84.2iT−5.04e3T2 1 + 84.2iT - 5.04e3T^{2}
73 1−63.5iT−5.32e3T2 1 - 63.5iT - 5.32e3T^{2}
79 1+(−9.06−15.7i)T+(−3.12e3+5.40e3i)T2 1 + (-9.06 - 15.7i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(−50.4−87.4i)T+(−3.44e3+5.96e3i)T2 1 + (-50.4 - 87.4i)T + (-3.44e3 + 5.96e3i)T^{2}
89 1−86.3iT−7.92e3T2 1 - 86.3iT - 7.92e3T^{2}
97 1+(59.7−34.5i)T+(4.70e3−8.14e3i)T2 1 + (59.7 - 34.5i)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

−15.42955400021147141409651807485, −14.37795540010183591004370900837, −13.39634089776840289928643080861, −12.46333931691205312137833912488, −10.79282436270445560020480082005, −9.570212570434718229995477318692, −7.59979246113986030873425338546, −6.32653310458610361001259408018, −5.69629086903218768537891853689, −2.32491768760797074064579614219, 3.00557186732055200284956037568, 4.57551088411597430169418342083, 6.33927323467611824228149438624, 8.452766205842448678441882654944, 10.02032034144418641537961358714, 10.59556015753417140654719445380, 12.22778319945829412963398452829, 13.17377282573798993649088154887, 14.25731778201918151919542591099, 15.97725831242608645657249393281

Graph of the ZZ-function along the critical line