Properties

Label 2-224-7.4-c1-0-5
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/2ΓC(s)L(s)=((0.198+0.980i)Λ(2s)\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/2ΓC(s+1/2)L(s)=((0.198+0.980i)Λ(1s)\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    =    2572^{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(193,)\chi_{224} (193, \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 0.2989900.365447i0.298990 - 0.365447i
L(12)L(\frac12) \approx 0.2989900.365447i0.298990 - 0.365447i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+(1.202.09i)T+(1.52.59i)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+(0.2070.358i)T+(5.59.52i)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+(8.514.7i)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 12.82T+29T2 1 - 2.82T + 29T^{2}
31 1+(4.207.28i)T+(15.526.8i)T2 1 + (4.20 - 7.28i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.322.30i)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+(26.545.8i)T2 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.44+7.70i)T+(29.551.0i)T2 1 + (-4.44 + 7.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.322.30i)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+(33.558.0i)T2 1 + (-5.62 + 9.73i)T + (-33.5 - 58.0i)T^{2}
71 12.34T+71T2 1 - 2.34T + 71T^{2}
73 1+(1.67+2.89i)T+(36.563.2i)T2 1 + (-1.67 + 2.89i)T + (-36.5 - 63.2i)T^{2}
79 1+(4.036.98i)T+(39.5+68.4i)T2 1 + (-4.03 - 6.98i)T + (-39.5 + 68.4i)T^{2}
83 115.3T+83T2 1 - 15.3T + 83T^{2}
89 1+(4.57.79i)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(1αj,pps)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

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

Graph of the ZZ-function along the critical line