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Properties

Label 2-28-28.3-c1-0-1
Degree 22
Conductor 2828
Sign 0.832+0.553i0.832 + 0.553i
Analytic cond. 0.2235810.223581
Root an. cond. 0.4728430.472843
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  + (0.366 − 1.36i)2-s + (−0.866 + 1.5i)3-s + (−1.73 − i)4-s + (−1.5 + 0.866i)5-s + (1.73 + 1.73i)6-s + (1.73 − 2i)7-s + (−2 + 1.99i)8-s + (0.633 + 2.36i)10-s + (−0.866 − 0.5i)11-s + (3 − 1.73i)12-s − 3.46i·13-s + (−2.09 − 3.09i)14-s − 3i·15-s + (1.99 + 3.46i)16-s + (−1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (−0.670 + 0.387i)5-s + (0.707 + 0.707i)6-s + (0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (0.200 + 0.748i)10-s + (−0.261 − 0.150i)11-s + (0.866 − 0.499i)12-s − 0.960i·13-s + (−0.560 − 0.827i)14-s − 0.774i·15-s + (0.499 + 0.866i)16-s + (−0.363 − 0.210i)17-s + ⋯

Functional equation

Λ(s)=(28s/2 ΓC(s) L(s)=((0.832+0.553i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(28s/2 ΓC(s+1/2) L(s)=((0.832+0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2828    =    22⋅72^{2} \cdot 7
Sign: 0.832+0.553i0.832 + 0.553i
Analytic conductor: 0.2235810.223581
Root analytic conductor: 0.4728430.472843
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ28(3,⋅)\chi_{28} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 28, ( :1/2), 0.832+0.553i)(2,\ 28,\ (\ :1/2),\ 0.832 + 0.553i)

Particular Values

L(1)L(1) ≈\approx 0.595521−0.179974i0.595521 - 0.179974i
L(12)L(\frac12) ≈\approx 0.595521−0.179974i0.595521 - 0.179974i
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+(−0.366+1.36i)T 1 + (-0.366 + 1.36i)T
7 1+(−1.73+2i)T 1 + (-1.73 + 2i)T
good3 1+(0.866−1.5i)T+(−1.5−2.59i)T2 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.5−0.866i)T+(2.5−4.33i)T2 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2}
11 1+(0.866+0.5i)T+(5.5+9.52i)T2 1 + (0.866 + 0.5i)T + (5.5 + 9.52i)T^{2}
13 1+3.46iT−13T2 1 + 3.46iT - 13T^{2}
17 1+(1.5+0.866i)T+(8.5+14.7i)T2 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2}
19 1+(−2.59−4.5i)T+(−9.5+16.4i)T2 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.866−0.5i)T+(11.5−19.9i)T2 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+(−0.866+1.5i)T+(−15.5−26.8i)T2 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.5+2.59i)T+(−18.5+32.0i)T2 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2}
41 1−3.46iT−41T2 1 - 3.46iT - 41T^{2}
43 1+2iT−43T2 1 + 2iT - 43T^{2}
47 1+(4.33+7.5i)T+(−23.5+40.7i)T2 1 + (4.33 + 7.5i)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+(−2.59+4.5i)T+(−29.5−51.0i)T2 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.5−2.59i)T+(30.5−52.8i)T2 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2}
67 1+(2.59+1.5i)T+(33.5+58.0i)T2 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2}
71 1−14iT−71T2 1 - 14iT - 71T^{2}
73 1+(−7.5−4.33i)T+(36.5+63.2i)T2 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2}
79 1+(7.79−4.5i)T+(39.5−68.4i)T2 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2}
83 1−13.8T+83T2 1 - 13.8T + 83T^{2}
89 1+(−13.5+7.79i)T+(44.5−77.0i)T2 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2}
97 1−17.3iT−97T2 1 - 17.3iT - 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

−17.32693724329760583976165417435, −15.87819465554784721739364562836, −14.71893569724865312437462693754, −13.40630963850504944178987305132, −11.76458414434093393874729517241, −10.81906248416036932328431871622, −10.02830839705813508541350777870, −7.948113451221559700796396614690, −5.21055972755172199076251286197, −3.79023935525388777841933541037, 4.72839043081946664752190802459, 6.41992193808807162362530336494, 7.74188899548957632199770281072, 9.025421267076234067385984904885, 11.67827119083028944990220808552, 12.43600940000912869275370860664, 13.76327518727332126738488530270, 15.18593175098594480682106124261, 16.10913315156939780936170302631, 17.52777971902752659332508962174

Graph of the ZZ-function along the critical line