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Properties

Label 2-45-45.4-c1-0-3
Degree 22
Conductor 4545
Sign 0.872+0.488i0.872 + 0.488i
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  + (1.67 − 0.965i)2-s + (−1.67 + 0.448i)3-s + (0.866 − 1.50i)4-s + (−2.09 + 0.792i)5-s + (−2.36 + 2.36i)6-s + (0.776 − 0.448i)7-s + 0.517i·8-s + (2.59 − 1.50i)9-s + (−2.73 + 3.34i)10-s + (−2.36 − 4.09i)11-s + (−0.776 + 2.89i)12-s + (2.12 + 1.22i)13-s + (0.866 − 1.50i)14-s + (3.14 − 2.26i)15-s + (2.23 + 3.86i)16-s + 0.378i·17-s + ⋯
L(s)  = 1  + (1.18 − 0.683i)2-s + (−0.965 + 0.258i)3-s + (0.433 − 0.750i)4-s + (−0.935 + 0.354i)5-s + (−0.965 + 0.965i)6-s + (0.293 − 0.169i)7-s + 0.183i·8-s + (0.866 − 0.5i)9-s + (−0.863 + 1.05i)10-s + (−0.713 − 1.23i)11-s + (−0.224 + 0.836i)12-s + (0.588 + 0.339i)13-s + (0.231 − 0.400i)14-s + (0.811 − 0.584i)15-s + (0.558 + 0.966i)16-s + 0.0919i·17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.872+0.488i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1/2) L(s)=((0.872+0.488i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.872+0.488i0.872 + 0.488i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ45(4,⋅)\chi_{45} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1/2), 0.872+0.488i)(2,\ 45,\ (\ :1/2),\ 0.872 + 0.488i)

Particular Values

L(1)L(1) ≈\approx 0.946520−0.246828i0.946520 - 0.246828i
L(12)L(\frac12) ≈\approx 0.946520−0.246828i0.946520 - 0.246828i
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+(1.67−0.448i)T 1 + (1.67 - 0.448i)T
5 1+(2.09−0.792i)T 1 + (2.09 - 0.792i)T
good2 1+(−1.67+0.965i)T+(1−1.73i)T2 1 + (-1.67 + 0.965i)T + (1 - 1.73i)T^{2}
7 1+(−0.776+0.448i)T+(3.5−6.06i)T2 1 + (-0.776 + 0.448i)T + (3.5 - 6.06i)T^{2}
11 1+(2.36+4.09i)T+(−5.5+9.52i)T2 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.12−1.22i)T+(6.5+11.2i)T2 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2}
17 1−0.378iT−17T2 1 - 0.378iT - 17T^{2}
19 1+2.73T+19T2 1 + 2.73T + 19T^{2}
23 1+(1.67+0.965i)T+(11.5+19.9i)T2 1 + (1.67 + 0.965i)T + (11.5 + 19.9i)T^{2}
29 1+(−3.23−5.59i)T+(−14.5+25.1i)T2 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.36+2.36i)T+(−15.5−26.8i)T2 1 + (-1.36 + 2.36i)T + (-15.5 - 26.8i)T^{2}
37 1+4.24iT−37T2 1 + 4.24iT - 37T^{2}
41 1+(2.13−3.69i)T+(−20.5−35.5i)T2 1 + (2.13 - 3.69i)T + (-20.5 - 35.5i)T^{2}
43 1+(7.91−4.57i)T+(21.5−37.2i)T2 1 + (7.91 - 4.57i)T + (21.5 - 37.2i)T^{2}
47 1+(−3.79+2.19i)T+(23.5−40.7i)T2 1 + (-3.79 + 2.19i)T + (23.5 - 40.7i)T^{2}
53 1+3.86iT−53T2 1 + 3.86iT - 53T^{2}
59 1+(−1.26+2.19i)T+(−29.5−51.0i)T2 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(5.33+9.23i)T+(−30.5+52.8i)T2 1 + (5.33 + 9.23i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.45−2.56i)T+(33.5+58.0i)T2 1 + (-4.45 - 2.56i)T + (33.5 + 58.0i)T^{2}
71 1−3.80T+71T2 1 - 3.80T + 71T^{2}
73 1+8.48iT−73T2 1 + 8.48iT - 73T^{2}
79 1+(−0.267−0.464i)T+(−39.5+68.4i)T2 1 + (-0.267 - 0.464i)T + (-39.5 + 68.4i)T^{2}
83 1+(−9.02+5.20i)T+(41.5−71.8i)T2 1 + (-9.02 + 5.20i)T + (41.5 - 71.8i)T^{2}
89 1−7.39T+89T2 1 - 7.39T + 89T^{2}
97 1+(8.90−5.13i)T+(48.5−84.0i)T2 1 + (8.90 - 5.13i)T + (48.5 - 84.0i)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

−15.68936037661310093880467917980, −14.50407941054231237811638291061, −13.26059582549936840659233151371, −12.16043112807290934132212929560, −11.17761564294142646094874816322, −10.69007699270294443289187580493, −8.235601823355759550425838099252, −6.29060068472210823633506581082, −4.82087524989054237548181658641, −3.53658687071640279818941773544, 4.29990823860780370590205816847, 5.29860820502979910190626681338, 6.77799752033855071011296975596, 7.951088943996130534695981124934, 10.28785459524726241120347842202, 11.80808462310258572700290070629, 12.55854724526281259284739526759, 13.51112654662852440764628839805, 15.22658234956860481282968740204, 15.59736085223624226346254995482

Graph of the ZZ-function along the critical line