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Properties

Label 2-1224-17.2-c1-0-21
Degree 22
Conductor 12241224
Sign −0.878+0.477i-0.878 + 0.477i
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  + (−1.10 − 2.66i)5-s + (1.25 − 3.03i)7-s + (−1.66 − 0.691i)11-s + 1.79i·13-s + (0.985 − 4.00i)17-s + (2.04 + 2.04i)19-s + (−2.97 − 1.23i)23-s + (−2.34 + 2.34i)25-s + (−2.39 − 5.78i)29-s + (−0.769 + 0.318i)31-s − 9.48·35-s + (−3.77 + 1.56i)37-s + (−3.11 + 7.51i)41-s + (−1.19 + 1.19i)43-s + 6.25i·47-s + ⋯
L(s)  = 1  + (−0.493 − 1.19i)5-s + (0.475 − 1.14i)7-s + (−0.503 − 0.208i)11-s + 0.497i·13-s + (0.238 − 0.971i)17-s + (0.468 + 0.468i)19-s + (−0.620 − 0.257i)23-s + (−0.468 + 0.468i)25-s + (−0.444 − 1.07i)29-s + (−0.138 + 0.0572i)31-s − 1.60·35-s + (−0.620 + 0.256i)37-s + (−0.486 + 1.17i)41-s + (−0.182 + 0.182i)43-s + 0.912i·47-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((−0.878+0.477i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s+1/2) L(s)=((−0.878+0.477i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: −0.878+0.477i-0.878 + 0.477i
Analytic conductor: 9.773689.77368
Root analytic conductor: 3.126293.12629
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1224(937,⋅)\chi_{1224} (937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :1/2), −0.878+0.477i)(2,\ 1224,\ (\ :1/2),\ -0.878 + 0.477i)

Particular Values

L(1)L(1) ≈\approx 1.0499950071.049995007
L(12)L(\frac12) ≈\approx 1.0499950071.049995007
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+(−0.985+4.00i)T 1 + (-0.985 + 4.00i)T
good5 1+(1.10+2.66i)T+(−3.53+3.53i)T2 1 + (1.10 + 2.66i)T + (-3.53 + 3.53i)T^{2}
7 1+(−1.25+3.03i)T+(−4.94−4.94i)T2 1 + (-1.25 + 3.03i)T + (-4.94 - 4.94i)T^{2}
11 1+(1.66+0.691i)T+(7.77+7.77i)T2 1 + (1.66 + 0.691i)T + (7.77 + 7.77i)T^{2}
13 1−1.79iT−13T2 1 - 1.79iT - 13T^{2}
19 1+(−2.04−2.04i)T+19iT2 1 + (-2.04 - 2.04i)T + 19iT^{2}
23 1+(2.97+1.23i)T+(16.2+16.2i)T2 1 + (2.97 + 1.23i)T + (16.2 + 16.2i)T^{2}
29 1+(2.39+5.78i)T+(−20.5+20.5i)T2 1 + (2.39 + 5.78i)T + (-20.5 + 20.5i)T^{2}
31 1+(0.769−0.318i)T+(21.9−21.9i)T2 1 + (0.769 - 0.318i)T + (21.9 - 21.9i)T^{2}
37 1+(3.77−1.56i)T+(26.1−26.1i)T2 1 + (3.77 - 1.56i)T + (26.1 - 26.1i)T^{2}
41 1+(3.11−7.51i)T+(−28.9−28.9i)T2 1 + (3.11 - 7.51i)T + (-28.9 - 28.9i)T^{2}
43 1+(1.19−1.19i)T−43iT2 1 + (1.19 - 1.19i)T - 43iT^{2}
47 1−6.25iT−47T2 1 - 6.25iT - 47T^{2}
53 1+(4.85+4.85i)T+53iT2 1 + (4.85 + 4.85i)T + 53iT^{2}
59 1+(4.81−4.81i)T−59iT2 1 + (4.81 - 4.81i)T - 59iT^{2}
61 1+(−5.35+12.9i)T+(−43.1−43.1i)T2 1 + (-5.35 + 12.9i)T + (-43.1 - 43.1i)T^{2}
67 1+13.3T+67T2 1 + 13.3T + 67T^{2}
71 1+(−9.25+3.83i)T+(50.2−50.2i)T2 1 + (-9.25 + 3.83i)T + (50.2 - 50.2i)T^{2}
73 1+(−0.877−2.11i)T+(−51.6+51.6i)T2 1 + (-0.877 - 2.11i)T + (-51.6 + 51.6i)T^{2}
79 1+(13.4+5.58i)T+(55.8+55.8i)T2 1 + (13.4 + 5.58i)T + (55.8 + 55.8i)T^{2}
83 1+(−8.07−8.07i)T+83iT2 1 + (-8.07 - 8.07i)T + 83iT^{2}
89 1+16.4iT−89T2 1 + 16.4iT - 89T^{2}
97 1+(6.34+15.3i)T+(−68.5+68.5i)T2 1 + (6.34 + 15.3i)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

−9.441468129929827733485195793544, −8.367404127201515643207568530670, −7.86838306133741647885595673442, −7.13565306481382791113935717208, −5.92401736985480213572031786813, −4.81516154621561880019784470502, −4.40696724222079769811523261313, −3.29047963124872771981006873554, −1.61926588041217776757982919474, −0.44506913901164339091540459362, 1.93159167772167607577653588116, 2.93462803799876286652693241000, 3.77755490573068693686716492857, 5.17810490099376225956911351469, 5.78739026141636165205973623702, 6.88291671017115834078319908206, 7.59616550236636423207985091654, 8.379386264799849365961901452300, 9.160982726052205153405372703817, 10.33401590623035804719633126121

Graph of the ZZ-function along the critical line