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Properties

Label 2-45-45.32-c1-0-3
Degree 22
Conductor 4545
Sign −0.455+0.890i-0.455 + 0.890i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 0.430i)2-s + (−1.08 − 1.35i)3-s + (0.661 + 0.382i)4-s + (−1.24 − 1.85i)5-s + (1.15 + 2.63i)6-s + (0.465 − 1.73i)7-s + (1.45 + 1.45i)8-s + (−0.661 + 2.92i)9-s + (1.20 + 3.51i)10-s + (3.12 − 1.80i)11-s + (−0.198 − 1.30i)12-s + (0.342 + 1.27i)13-s + (−1.49 + 2.59i)14-s + (−1.15 + 3.69i)15-s + (−2.47 − 4.28i)16-s + (−0.277 + 0.277i)17-s + ⋯
L(s)  = 1  + (−1.13 − 0.304i)2-s + (−0.624 − 0.781i)3-s + (0.330 + 0.191i)4-s + (−0.558 − 0.829i)5-s + (0.471 + 1.07i)6-s + (0.175 − 0.656i)7-s + (0.513 + 0.513i)8-s + (−0.220 + 0.975i)9-s + (0.381 + 1.11i)10-s + (0.942 − 0.544i)11-s + (−0.0573 − 0.377i)12-s + (0.0950 + 0.354i)13-s + (−0.399 + 0.692i)14-s + (−0.299 + 0.954i)15-s + (−0.618 − 1.07i)16-s + (−0.0671 + 0.0671i)17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((−0.455+0.890i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1/2) L(s)=((−0.455+0.890i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: −0.455+0.890i-0.455 + 0.890i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ45(32,⋅)\chi_{45} (32, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1/2), −0.455+0.890i)(2,\ 45,\ (\ :1/2),\ -0.455 + 0.890i)

Particular Values

L(1)L(1) ≈\approx 0.189266−0.309287i0.189266 - 0.309287i
L(12)L(\frac12) ≈\approx 0.189266−0.309287i0.189266 - 0.309287i
L(32)L(\frac{3}{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+(1.08+1.35i)T 1 + (1.08 + 1.35i)T
5 1+(1.24+1.85i)T 1 + (1.24 + 1.85i)T
good2 1+(1.60+0.430i)T+(1.73+i)T2 1 + (1.60 + 0.430i)T + (1.73 + i)T^{2}
7 1+(−0.465+1.73i)T+(−6.06−3.5i)T2 1 + (-0.465 + 1.73i)T + (-6.06 - 3.5i)T^{2}
11 1+(−3.12+1.80i)T+(5.5−9.52i)T2 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2}
13 1+(−0.342−1.27i)T+(−11.2+6.5i)T2 1 + (-0.342 - 1.27i)T + (-11.2 + 6.5i)T^{2}
17 1+(0.277−0.277i)T−17iT2 1 + (0.277 - 0.277i)T - 17iT^{2}
19 1+6.25iT−19T2 1 + 6.25iT - 19T^{2}
23 1+(−2.16+0.579i)T+(19.9−11.5i)T2 1 + (-2.16 + 0.579i)T + (19.9 - 11.5i)T^{2}
29 1+(−1.56−2.71i)T+(−14.5+25.1i)T2 1 + (-1.56 - 2.71i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.42−4.20i)T+(−15.5−26.8i)T2 1 + (2.42 - 4.20i)T + (-15.5 - 26.8i)T^{2}
37 1+(−5.55−5.55i)T+37iT2 1 + (-5.55 - 5.55i)T + 37iT^{2}
41 1+(−1.29−0.744i)T+(20.5+35.5i)T2 1 + (-1.29 - 0.744i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.10−1.10i)T+(37.2+21.5i)T2 1 + (-4.10 - 1.10i)T + (37.2 + 21.5i)T^{2}
47 1+(3.82+1.02i)T+(40.7+23.5i)T2 1 + (3.82 + 1.02i)T + (40.7 + 23.5i)T^{2}
53 1+(7.48+7.48i)T+53iT2 1 + (7.48 + 7.48i)T + 53iT^{2}
59 1+(0.279−0.483i)T+(−29.5−51.0i)T2 1 + (0.279 - 0.483i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.96+5.13i)T+(−30.5+52.8i)T2 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2}
67 1+(−10.8+2.90i)T+(58.0−33.5i)T2 1 + (-10.8 + 2.90i)T + (58.0 - 33.5i)T^{2}
71 1−8.01iT−71T2 1 - 8.01iT - 71T^{2}
73 1+(1.29−1.29i)T−73iT2 1 + (1.29 - 1.29i)T - 73iT^{2}
79 1+(6.96−4.02i)T+(39.5−68.4i)T2 1 + (6.96 - 4.02i)T + (39.5 - 68.4i)T^{2}
83 1+(0.150−0.560i)T+(−71.8−41.5i)T2 1 + (0.150 - 0.560i)T + (-71.8 - 41.5i)T^{2}
89 1−16.4T+89T2 1 - 16.4T + 89T^{2}
97 1+(1.37−5.14i)T+(−84.0−48.5i)T2 1 + (1.37 - 5.14i)T + (-84.0 - 48.5i)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

−16.16550312091032499564447683183, −14.12787949575164915846621363109, −13.01599154683985898718344117318, −11.59618456214797386683551588260, −10.94373834891778675237181115080, −9.217580658652879587567557510424, −8.188763433535154635786609336300, −6.91173020135586799611166266099, −4.79838109536486840984948802532, −1.05222706323018633244862708778, 4.03074218264799297799391964867, 6.19018051018682540099364219384, 7.66167354741088512257800433577, 9.097425269889251913865302158348, 10.10714816390662001385579190722, 11.21034834923277374846623315088, 12.33813094589998815019163773622, 14.53344572241206494435041750329, 15.39920808406335687945735858078, 16.38834156846371629903760287786

Graph of the ZZ-function along the critical line