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Properties

Label 2-1320-1320.173-c0-0-1
Degree 22
Conductor 13201320
Sign −0.442−0.896i-0.442 - 0.896i
Analytic cond. 0.6587650.658765
Root an. cond. 0.8116430.811643
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 + 0.156i)2-s + (−0.453 + 0.891i)3-s + (0.951 + 0.309i)4-s + (0.453 + 0.891i)5-s + (−0.587 + 0.809i)6-s + (−1.76 + 0.896i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (0.309 + 0.951i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−1.87 + 0.610i)14-s − 1.00·15-s + (0.809 + 0.587i)16-s + (−0.453 − 0.891i)18-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (−0.453 + 0.891i)3-s + (0.951 + 0.309i)4-s + (0.453 + 0.891i)5-s + (−0.587 + 0.809i)6-s + (−1.76 + 0.896i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (0.309 + 0.951i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−1.87 + 0.610i)14-s − 1.00·15-s + (0.809 + 0.587i)16-s + (−0.453 − 0.891i)18-s + ⋯

Functional equation

Λ(s)=(1320s/2 ΓC(s) L(s)=((−0.442−0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1320s/2 ΓC(s) L(s)=((−0.442−0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13201320    =    23⋅3⋅5⋅112^{3} \cdot 3 \cdot 5 \cdot 11
Sign: −0.442−0.896i-0.442 - 0.896i
Analytic conductor: 0.6587650.658765
Root analytic conductor: 0.8116430.811643
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1320(173,⋅)\chi_{1320} (173, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1320, ( :0), −0.442−0.896i)(2,\ 1320,\ (\ :0),\ -0.442 - 0.896i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5601654641.560165464
L(12)L(\frac12) ≈\approx 1.5601654641.560165464
L(1)L(1) 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+(−0.987−0.156i)T 1 + (-0.987 - 0.156i)T
3 1+(0.453−0.891i)T 1 + (0.453 - 0.891i)T
5 1+(−0.453−0.891i)T 1 + (-0.453 - 0.891i)T
11 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1+(1.76−0.896i)T+(0.587−0.809i)T2 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2}
13 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
17 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1−iT2 1 - iT^{2}
29 1+(−0.280+0.863i)T+(−0.809−0.587i)T2 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2}
31 1+(0.951−0.690i)T+(0.309−0.951i)T2 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2}
37 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
41 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
53 1+(−1.59−0.253i)T+(0.951+0.309i)T2 1 + (-1.59 - 0.253i)T + (0.951 + 0.309i)T^{2}
59 1+(−0.297−0.0966i)T+(0.809+0.587i)T2 1 + (-0.297 - 0.0966i)T + (0.809 + 0.587i)T^{2}
61 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
67 1+iT2 1 + iT^{2}
71 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
73 1+(0.809+1.58i)T+(−0.587+0.809i)T2 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2}
79 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
83 1+(−0.610+0.0966i)T+(0.951−0.309i)T2 1 + (-0.610 + 0.0966i)T + (0.951 - 0.309i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.278−1.76i)T+(−0.951−0.309i)T2 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)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

−10.21318972340847034020890940279, −9.437202644754023649675549918229, −8.725948124830310537024524316551, −7.16437776054527644806100217752, −6.30600975392338890198525903251, −6.07067171417993449585737642945, −5.26454574345341886525984975955, −3.83720917819246181384200349883, −3.32015434295935359615211180522, −2.49182309999008282934709298529, 1.05410395787597964012478253207, 2.27446277198320493719835945664, 3.54492206022855051899593080813, 4.44268015443659088186460750946, 5.51505068281514958169530478183, 6.22298217024864011537210189550, 6.91745825198739960847028094300, 7.44073224271043118988215446029, 8.840340843444649151446490123065, 9.805313314455003515468603795635

Graph of the ZZ-function along the critical line