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Properties

Label 2-224-7.2-c1-0-0
Degree 22
Conductor 224224
Sign −0.198−0.980i-0.198 - 0.980i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
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  + (−1.20 − 2.09i)3-s + (−1.91 + 3.31i)5-s + (−1 + 2.44i)7-s + (−1.41 + 2.44i)9-s + (−0.207 − 0.358i)11-s − 2.82·13-s + 9.24·15-s + (2.91 + 5.04i)17-s + (−1.79 + 3.10i)19-s + (6.32 − 0.866i)21-s + (−1.62 + 2.80i)23-s + (−4.82 − 8.36i)25-s − 0.414·27-s + 2.82·29-s + (−4.20 − 7.28i)31-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)3-s + (−0.856 + 1.48i)5-s + (−0.377 + 0.925i)7-s + (−0.471 + 0.816i)9-s + (−0.0624 − 0.108i)11-s − 0.784·13-s + 2.38·15-s + (0.706 + 1.22i)17-s + (−0.411 + 0.712i)19-s + (1.38 − 0.188i)21-s + (−0.338 + 0.585i)23-s + (−0.965 − 1.67i)25-s − 0.0797·27-s + 0.525·29-s + (−0.755 − 1.30i)31-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((−0.198−0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((−0.198−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: −0.198−0.980i-0.198 - 0.980i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(65,⋅)\chi_{224} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), −0.198−0.980i)(2,\ 224,\ (\ :1/2),\ -0.198 - 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.298990+0.365447i0.298990 + 0.365447i
L(12)L(\frac12) ≈\approx 0.298990+0.365447i0.298990 + 0.365447i
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
7 1+(1−2.44i)T 1 + (1 - 2.44i)T
good3 1+(1.20+2.09i)T+(−1.5+2.59i)T2 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.91−3.31i)T+(−2.5−4.33i)T2 1 + (1.91 - 3.31i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.207+0.358i)T+(−5.5+9.52i)T2 1 + (0.207 + 0.358i)T + (-5.5 + 9.52i)T^{2}
13 1+2.82T+13T2 1 + 2.82T + 13T^{2}
17 1+(−2.91−5.04i)T+(−8.5+14.7i)T2 1 + (-2.91 - 5.04i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.79−3.10i)T+(−9.5−16.4i)T2 1 + (1.79 - 3.10i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.62−2.80i)T+(−11.5−19.9i)T2 1 + (1.62 - 2.80i)T + (-11.5 - 19.9i)T^{2}
29 1−2.82T+29T2 1 - 2.82T + 29T^{2}
31 1+(4.20+7.28i)T+(−15.5+26.8i)T2 1 + (4.20 + 7.28i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.32+2.30i)T+(−18.5−32.0i)T2 1 + (-1.32 + 2.30i)T + (-18.5 - 32.0i)T^{2}
41 1+1.17T+41T2 1 + 1.17T + 41T^{2}
43 1+1.65T+43T2 1 + 1.65T + 43T^{2}
47 1+(3.79−6.56i)T+(−23.5−40.7i)T2 1 + (3.79 - 6.56i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.5−0.866i)T+(−26.5+45.8i)T2 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.44−7.70i)T+(−29.5+51.0i)T2 1 + (-4.44 - 7.70i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.32+2.30i)T+(−30.5−52.8i)T2 1 + (-1.32 + 2.30i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.62−9.73i)T+(−33.5+58.0i)T2 1 + (-5.62 - 9.73i)T + (-33.5 + 58.0i)T^{2}
71 1−2.34T+71T2 1 - 2.34T + 71T^{2}
73 1+(−1.67−2.89i)T+(−36.5+63.2i)T2 1 + (-1.67 - 2.89i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.03+6.98i)T+(−39.5−68.4i)T2 1 + (-4.03 + 6.98i)T + (-39.5 - 68.4i)T^{2}
83 1−15.3T+83T2 1 - 15.3T + 83T^{2}
89 1+(−4.5+7.79i)T+(−44.5−77.0i)T2 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2}
97 1+6.82T+97T2 1 + 6.82T + 97T^{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

−12.28018101645004293635632017827, −11.78212290915825815982425658793, −10.87432782568891830810023882521, −9.831682518992228136269149994960, −8.110494502140131984945290582221, −7.43686613639004446020035357095, −6.42665345028336530833284235642, −5.76986169724734378208272622338, −3.65352756788188829622888871893, −2.25281250701362989151535054890, 0.41493240246859933979110567976, 3.63476503124581709168269301517, 4.74970796429945316308409696946, 5.08899489121680019822125765024, 6.94125238705006727806446586258, 8.115216830677667084266445815261, 9.294811313398675576747381963330, 9.998302211544781354820083614034, 11.00470647595595192609309142053, 11.95447090380714967130093443241

Graph of the ZZ-function along the critical line