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Properties

Label 2-450-225.79-c1-0-11
Degree 22
Conductor 450450
Sign 0.713−0.700i0.713 - 0.700i
Analytic cond. 3.593263.59326
Root an. cond. 1.895591.89559
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  + (0.743 − 0.669i)2-s + (0.679 + 1.59i)3-s + (0.104 − 0.994i)4-s + (0.798 + 2.08i)5-s + (1.57 + 0.728i)6-s + (2.11 + 1.21i)7-s + (−0.587 − 0.809i)8-s + (−2.07 + 2.16i)9-s + (1.99 + 1.01i)10-s + (0.211 + 0.234i)11-s + (1.65 − 0.509i)12-s + (−4.38 − 3.95i)13-s + (2.38 − 0.507i)14-s + (−2.78 + 2.69i)15-s + (−0.978 − 0.207i)16-s + (3.64 + 5.01i)17-s + ⋯
L(s)  = 1  + (0.525 − 0.473i)2-s + (0.392 + 0.919i)3-s + (0.0522 − 0.497i)4-s + (0.356 + 0.934i)5-s + (0.641 + 0.297i)6-s + (0.798 + 0.460i)7-s + (−0.207 − 0.286i)8-s + (−0.691 + 0.722i)9-s + (0.629 + 0.322i)10-s + (0.0637 + 0.0708i)11-s + (0.477 − 0.147i)12-s + (−1.21 − 1.09i)13-s + (0.637 − 0.135i)14-s + (−0.719 + 0.694i)15-s + (−0.244 − 0.0519i)16-s + (0.883 + 1.21i)17-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=((0.713−0.700i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+1/2) L(s)=((0.713−0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: 0.713−0.700i0.713 - 0.700i
Analytic conductor: 3.593263.59326
Root analytic conductor: 1.895591.89559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ450(79,⋅)\chi_{450} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 450, ( :1/2), 0.713−0.700i)(2,\ 450,\ (\ :1/2),\ 0.713 - 0.700i)

Particular Values

L(1)L(1) ≈\approx 2.05295+0.838880i2.05295 + 0.838880i
L(12)L(\frac12) ≈\approx 2.05295+0.838880i2.05295 + 0.838880i
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+(−0.743+0.669i)T 1 + (-0.743 + 0.669i)T
3 1+(−0.679−1.59i)T 1 + (-0.679 - 1.59i)T
5 1+(−0.798−2.08i)T 1 + (-0.798 - 2.08i)T
good7 1+(−2.11−1.21i)T+(3.5+6.06i)T2 1 + (-2.11 - 1.21i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.211−0.234i)T+(−1.14+10.9i)T2 1 + (-0.211 - 0.234i)T + (-1.14 + 10.9i)T^{2}
13 1+(4.38+3.95i)T+(1.35+12.9i)T2 1 + (4.38 + 3.95i)T + (1.35 + 12.9i)T^{2}
17 1+(−3.64−5.01i)T+(−5.25+16.1i)T2 1 + (-3.64 - 5.01i)T + (-5.25 + 16.1i)T^{2}
19 1+(0.706−0.513i)T+(5.87−18.0i)T2 1 + (0.706 - 0.513i)T + (5.87 - 18.0i)T^{2}
23 1+(1.01+4.78i)T+(−21.0+9.35i)T2 1 + (1.01 + 4.78i)T + (-21.0 + 9.35i)T^{2}
29 1+(−8.57+3.81i)T+(19.4−21.5i)T2 1 + (-8.57 + 3.81i)T + (19.4 - 21.5i)T^{2}
31 1+(−2.30−1.02i)T+(20.7+23.0i)T2 1 + (-2.30 - 1.02i)T + (20.7 + 23.0i)T^{2}
37 1+(−3.71+1.20i)T+(29.9−21.7i)T2 1 + (-3.71 + 1.20i)T + (29.9 - 21.7i)T^{2}
41 1+(−3.44+3.82i)T+(−4.28−40.7i)T2 1 + (-3.44 + 3.82i)T + (-4.28 - 40.7i)T^{2}
43 1+(−2.81−1.62i)T+(21.5+37.2i)T2 1 + (-2.81 - 1.62i)T + (21.5 + 37.2i)T^{2}
47 1+(3.16+7.11i)T+(−31.4+34.9i)T2 1 + (3.16 + 7.11i)T + (-31.4 + 34.9i)T^{2}
53 1+(3.18−4.38i)T+(−16.3−50.4i)T2 1 + (3.18 - 4.38i)T + (-16.3 - 50.4i)T^{2}
59 1+(−1.70+1.88i)T+(−6.16−58.6i)T2 1 + (-1.70 + 1.88i)T + (-6.16 - 58.6i)T^{2}
61 1+(6.20+6.89i)T+(−6.37+60.6i)T2 1 + (6.20 + 6.89i)T + (-6.37 + 60.6i)T^{2}
67 1+(0.119−0.269i)T+(−44.8−49.7i)T2 1 + (0.119 - 0.269i)T + (-44.8 - 49.7i)T^{2}
71 1+(−11.2−8.18i)T+(21.9+67.5i)T2 1 + (-11.2 - 8.18i)T + (21.9 + 67.5i)T^{2}
73 1+(8.80+2.86i)T+(59.0+42.9i)T2 1 + (8.80 + 2.86i)T + (59.0 + 42.9i)T^{2}
79 1+(13.4−5.96i)T+(52.8−58.7i)T2 1 + (13.4 - 5.96i)T + (52.8 - 58.7i)T^{2}
83 1+(1.24−0.130i)T+(81.1−17.2i)T2 1 + (1.24 - 0.130i)T + (81.1 - 17.2i)T^{2}
89 1+(−4.73+14.5i)T+(−72.0−52.3i)T2 1 + (-4.73 + 14.5i)T + (-72.0 - 52.3i)T^{2}
97 1+(1.09+2.44i)T+(−64.9+72.0i)T2 1 + (1.09 + 2.44i)T + (-64.9 + 72.0i)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

−11.02757259039552214224909813743, −10.20060557737958863028618909122, −9.959281807068018503076786795321, −8.509294431881055518779412388356, −7.72692954043390994934178759468, −6.18282171533770455544004618685, −5.31029620032885864998092748001, −4.33048661135735954980737606994, −3.06306350819034172171287044336, −2.25644677202684363097321546988, 1.31116224589345630609373135940, 2.72571207624468589274365620994, 4.42020004433785534925653482106, 5.16871816706686938927864086260, 6.36921547815263432943550025901, 7.40457666078113313115419060963, 7.947565948954483088615346867833, 9.023265781385794477046975438015, 9.770284554731779478800780064550, 11.48944056630536768118929536744

Graph of the ZZ-function along the critical line