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Properties

Label 2-45-45.29-c2-0-8
Degree 22
Conductor 4545
Sign 0.0797+0.996i0.0797 + 0.996i
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  + (1.14 − 1.98i)2-s + (−0.182 − 2.99i)3-s + (−0.629 − 1.09i)4-s + (0.662 + 4.95i)5-s + (−6.15 − 3.07i)6-s + (−6.04 − 3.49i)7-s + 6.28·8-s + (−8.93 + 1.09i)9-s + (10.6 + 4.36i)10-s + (15.8 + 9.12i)11-s + (−3.15 + 2.08i)12-s + (−3.66 + 2.11i)13-s + (−13.8 + 8.00i)14-s + (14.7 − 2.88i)15-s + (9.72 − 16.8i)16-s − 17.3·17-s + ⋯
L(s)  = 1  + (0.573 − 0.993i)2-s + (−0.0607 − 0.998i)3-s + (−0.157 − 0.272i)4-s + (0.132 + 0.991i)5-s + (−1.02 − 0.511i)6-s + (−0.863 − 0.498i)7-s + 0.785·8-s + (−0.992 + 0.121i)9-s + (1.06 + 0.436i)10-s + (1.43 + 0.829i)11-s + (−0.262 + 0.173i)12-s + (−0.282 + 0.162i)13-s + (−0.990 + 0.571i)14-s + (0.981 − 0.192i)15-s + (0.607 − 1.05i)16-s − 1.01·17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.0797+0.996i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0797 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1) L(s)=((0.0797+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0797 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.0797+0.996i0.0797 + 0.996i
Analytic conductor: 1.226161.22616
Root analytic conductor: 1.107321.10732
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ45(29,⋅)\chi_{45} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1), 0.0797+0.996i)(2,\ 45,\ (\ :1),\ 0.0797 + 0.996i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.03825−0.958541i1.03825 - 0.958541i
L(12)L(\frac12) ≈\approx 1.03825−0.958541i1.03825 - 0.958541i
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+(0.182+2.99i)T 1 + (0.182 + 2.99i)T
5 1+(−0.662−4.95i)T 1 + (-0.662 - 4.95i)T
good2 1+(−1.14+1.98i)T+(−2−3.46i)T2 1 + (-1.14 + 1.98i)T + (-2 - 3.46i)T^{2}
7 1+(6.04+3.49i)T+(24.5+42.4i)T2 1 + (6.04 + 3.49i)T + (24.5 + 42.4i)T^{2}
11 1+(−15.8−9.12i)T+(60.5+104.i)T2 1 + (-15.8 - 9.12i)T + (60.5 + 104. i)T^{2}
13 1+(3.66−2.11i)T+(84.5−146.i)T2 1 + (3.66 - 2.11i)T + (84.5 - 146. i)T^{2}
17 1+17.3T+289T2 1 + 17.3T + 289T^{2}
19 1+3.96T+361T2 1 + 3.96T + 361T^{2}
23 1+(−0.287−0.498i)T+(−264.5+458.i)T2 1 + (-0.287 - 0.498i)T + (-264.5 + 458. i)T^{2}
29 1+(18.1+10.4i)T+(420.5+728.i)T2 1 + (18.1 + 10.4i)T + (420.5 + 728. i)T^{2}
31 1+(16.7+28.9i)T+(−480.5+832.i)T2 1 + (16.7 + 28.9i)T + (-480.5 + 832. i)T^{2}
37 1−21.4iT−1.36e3T2 1 - 21.4iT - 1.36e3T^{2}
41 1+(−44.1+25.4i)T+(840.5−1.45e3i)T2 1 + (-44.1 + 25.4i)T + (840.5 - 1.45e3i)T^{2}
43 1+(7.15+4.13i)T+(924.5+1.60e3i)T2 1 + (7.15 + 4.13i)T + (924.5 + 1.60e3i)T^{2}
47 1+(7.57−13.1i)T+(−1.10e3−1.91e3i)T2 1 + (7.57 - 13.1i)T + (-1.10e3 - 1.91e3i)T^{2}
53 1−24.5T+2.80e3T2 1 - 24.5T + 2.80e3T^{2}
59 1+(−43.1+24.9i)T+(1.74e3−3.01e3i)T2 1 + (-43.1 + 24.9i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(31.4−54.5i)T+(−1.86e3−3.22e3i)T2 1 + (31.4 - 54.5i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−103.+59.7i)T+(2.24e3−3.88e3i)T2 1 + (-103. + 59.7i)T + (2.24e3 - 3.88e3i)T^{2}
71 1−66.8iT−5.04e3T2 1 - 66.8iT - 5.04e3T^{2}
73 1+48.9iT−5.32e3T2 1 + 48.9iT - 5.32e3T^{2}
79 1+(−58.9+102.i)T+(−3.12e3−5.40e3i)T2 1 + (-58.9 + 102. i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−3.66+6.34i)T+(−3.44e3−5.96e3i)T2 1 + (-3.66 + 6.34i)T + (-3.44e3 - 5.96e3i)T^{2}
89 1−100.iT−7.92e3T2 1 - 100. iT - 7.92e3T^{2}
97 1+(3.59+2.07i)T+(4.70e3+8.14e3i)T2 1 + (3.59 + 2.07i)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

−14.83536749203996603097053328467, −13.79819686660702695683996649980, −12.94675891028570479088422486010, −11.88434090759310527072376920214, −10.98739583367702718775557401462, −9.581696439629295290409110338640, −7.32877492292275247049451840253, −6.50164658672333376597082494318, −3.85005594779752640872282870343, −2.20289567176270860102901922860, 4.02694104350437408879623101922, 5.42289191754944835637019237260, 6.46214329264563744390051514449, 8.641846356701150683447544518937, 9.514069041515805454718697090701, 11.15626058296340910717159468686, 12.65569011986903993669024678010, 13.91441762562648086121266426401, 14.92939774509834665609479529192, 16.00512457137201257526767432009

Graph of the ZZ-function along the critical line