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Properties

Label 2-1224-17.9-c1-0-14
Degree 22
Conductor 12241224
Sign 0.892+0.451i0.892 + 0.451i
Analytic cond. 9.773689.77368
Root an. cond. 3.126293.12629
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  + (1.35 − 3.26i)5-s + (0.666 + 1.60i)7-s + (3.36 − 1.39i)11-s + 5.09i·13-s + (1.10 + 3.97i)17-s + (2.23 − 2.23i)19-s + (5.29 − 2.19i)23-s + (−5.30 − 5.30i)25-s + (−2.25 + 5.45i)29-s + (3.58 + 1.48i)31-s + 6.15·35-s + (0.0611 + 0.0253i)37-s + (−3.97 − 9.58i)41-s + (1.89 + 1.89i)43-s − 11.4i·47-s + ⋯
L(s)  = 1  + (0.605 − 1.46i)5-s + (0.251 + 0.608i)7-s + (1.01 − 0.420i)11-s + 1.41i·13-s + (0.266 + 0.963i)17-s + (0.512 − 0.512i)19-s + (1.10 − 0.457i)23-s + (−1.06 − 1.06i)25-s + (−0.419 + 1.01i)29-s + (0.644 + 0.267i)31-s + 1.04·35-s + (0.0100 + 0.00416i)37-s + (−0.620 − 1.49i)41-s + (0.289 + 0.289i)43-s − 1.66i·47-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((0.892+0.451i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s+1/2) L(s)=((0.892+0.451i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: 0.892+0.451i0.892 + 0.451i
Analytic conductor: 9.773689.77368
Root analytic conductor: 3.126293.12629
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1224(145,⋅)\chi_{1224} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :1/2), 0.892+0.451i)(2,\ 1224,\ (\ :1/2),\ 0.892 + 0.451i)

Particular Values

L(1)L(1) ≈\approx 2.0736578422.073657842
L(12)L(\frac12) ≈\approx 2.0736578422.073657842
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)
bad2 1 1
3 1 1
17 1+(−1.10−3.97i)T 1 + (-1.10 - 3.97i)T
good5 1+(−1.35+3.26i)T+(−3.53−3.53i)T2 1 + (-1.35 + 3.26i)T + (-3.53 - 3.53i)T^{2}
7 1+(−0.666−1.60i)T+(−4.94+4.94i)T2 1 + (-0.666 - 1.60i)T + (-4.94 + 4.94i)T^{2}
11 1+(−3.36+1.39i)T+(7.77−7.77i)T2 1 + (-3.36 + 1.39i)T + (7.77 - 7.77i)T^{2}
13 1−5.09iT−13T2 1 - 5.09iT - 13T^{2}
19 1+(−2.23+2.23i)T−19iT2 1 + (-2.23 + 2.23i)T - 19iT^{2}
23 1+(−5.29+2.19i)T+(16.2−16.2i)T2 1 + (-5.29 + 2.19i)T + (16.2 - 16.2i)T^{2}
29 1+(2.25−5.45i)T+(−20.5−20.5i)T2 1 + (2.25 - 5.45i)T + (-20.5 - 20.5i)T^{2}
31 1+(−3.58−1.48i)T+(21.9+21.9i)T2 1 + (-3.58 - 1.48i)T + (21.9 + 21.9i)T^{2}
37 1+(−0.0611−0.0253i)T+(26.1+26.1i)T2 1 + (-0.0611 - 0.0253i)T + (26.1 + 26.1i)T^{2}
41 1+(3.97+9.58i)T+(−28.9+28.9i)T2 1 + (3.97 + 9.58i)T + (-28.9 + 28.9i)T^{2}
43 1+(−1.89−1.89i)T+43iT2 1 + (-1.89 - 1.89i)T + 43iT^{2}
47 1+11.4iT−47T2 1 + 11.4iT - 47T^{2}
53 1+(2.59−2.59i)T−53iT2 1 + (2.59 - 2.59i)T - 53iT^{2}
59 1+(2.93+2.93i)T+59iT2 1 + (2.93 + 2.93i)T + 59iT^{2}
61 1+(1.53+3.69i)T+(−43.1+43.1i)T2 1 + (1.53 + 3.69i)T + (-43.1 + 43.1i)T^{2}
67 1+8.84T+67T2 1 + 8.84T + 67T^{2}
71 1+(1.37+0.569i)T+(50.2+50.2i)T2 1 + (1.37 + 0.569i)T + (50.2 + 50.2i)T^{2}
73 1+(3.83−9.26i)T+(−51.6−51.6i)T2 1 + (3.83 - 9.26i)T + (-51.6 - 51.6i)T^{2}
79 1+(−13.1+5.44i)T+(55.8−55.8i)T2 1 + (-13.1 + 5.44i)T + (55.8 - 55.8i)T^{2}
83 1+(2.07−2.07i)T−83iT2 1 + (2.07 - 2.07i)T - 83iT^{2}
89 1+2.32iT−89T2 1 + 2.32iT - 89T^{2}
97 1+(2.22−5.36i)T+(−68.5−68.5i)T2 1 + (2.22 - 5.36i)T + (-68.5 - 68.5i)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

−9.227986897390371410370244321908, −8.967327809589323334176818578077, −8.463873697655531110389576632554, −7.08051873195725505913828589555, −6.24230136504917328514397497383, −5.32944233895467335752976256205, −4.67307482797752877352524813383, −3.63195282592006708154928865669, −2.00928312257971940617224437888, −1.16427421841640735609977760857, 1.23032400032903346941640626804, 2.73152174446946201420608810293, 3.38290595757971699215656096957, 4.62603598919339874723225688583, 5.77106957311642162879621299695, 6.47675892592634765113797225485, 7.40016555638113715114252503226, 7.78655219541385471388096239832, 9.261987594558129264576760646133, 9.877339461562861794603390445886

Graph of the ZZ-function along the critical line