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Properties

Label 2-1320-1320.413-c0-0-3
Degree 22
Conductor 13201320
Sign −0.995+0.0965i-0.995 + 0.0965i
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  + (0.453 − 0.891i)2-s + (−0.156 − 0.987i)3-s + (−0.587 − 0.809i)4-s + (0.156 − 0.987i)5-s + (−0.951 − 0.309i)6-s + (0.896 + 0.142i)7-s + (−0.987 + 0.156i)8-s + (−0.951 + 0.309i)9-s + (−0.809 − 0.587i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.533 − 0.734i)14-s − 15-s + (−0.309 + 0.951i)16-s + (−0.156 + 0.987i)18-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.156 − 0.987i)3-s + (−0.587 − 0.809i)4-s + (0.156 − 0.987i)5-s + (−0.951 − 0.309i)6-s + (0.896 + 0.142i)7-s + (−0.987 + 0.156i)8-s + (−0.951 + 0.309i)9-s + (−0.809 − 0.587i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.533 − 0.734i)14-s − 15-s + (−0.309 + 0.951i)16-s + (−0.156 + 0.987i)18-s + ⋯

Functional equation

Λ(s)=(1320s/2 ΓC(s) L(s)=((−0.995+0.0965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1320s/2 ΓC(s) L(s)=((−0.995+0.0965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13201320    =    23⋅3⋅5⋅112^{3} \cdot 3 \cdot 5 \cdot 11
Sign: −0.995+0.0965i-0.995 + 0.0965i
Analytic conductor: 0.6587650.658765
Root analytic conductor: 0.8116430.811643
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1320(413,⋅)\chi_{1320} (413, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1320, ( :0), −0.995+0.0965i)(2,\ 1320,\ (\ :0),\ -0.995 + 0.0965i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2938973521.293897352
L(12)L(\frac12) ≈\approx 1.2938973521.293897352
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+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
3 1+(0.156+0.987i)T 1 + (0.156 + 0.987i)T
5 1+(−0.156+0.987i)T 1 + (-0.156 + 0.987i)T
11 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1+(−0.896−0.142i)T+(0.951+0.309i)T2 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2}
13 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
17 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
19 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
23 1−iT2 1 - iT^{2}
29 1+(0.253−0.183i)T+(0.309−0.951i)T2 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2}
31 1+(−0.587−1.80i)T+(−0.809+0.587i)T2 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2}
37 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
41 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
53 1+(0.280−0.550i)T+(−0.587−0.809i)T2 1 + (0.280 - 0.550i)T + (-0.587 - 0.809i)T^{2}
59 1+(−1.04−1.44i)T+(−0.309+0.951i)T2 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2}
61 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
67 1+iT2 1 + iT^{2}
71 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
73 1+(−0.309+1.95i)T+(−0.951−0.309i)T2 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2}
79 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
83 1+(0.734+1.44i)T+(−0.587+0.809i)T2 1 + (0.734 + 1.44i)T + (-0.587 + 0.809i)T^{2}
89 1+T2 1 + T^{2}
97 1+(1.76+0.896i)T+(0.587+0.809i)T2 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)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

−9.304043539756123159121813115712, −8.643946271870561136163212080382, −8.128127597756525706585917678597, −6.82370383503027853585408188821, −5.84614363120418028573579428899, −5.21535077798277189477024831855, −4.34893624421944344836003945311, −3.08699768363626840554484186772, −1.80507312353087248848677387087, −1.09072612856837774498989076706, 2.41048268212136326529470636377, 3.66853739311702328132848032959, 4.30268763872519183977617838696, 5.18042097097884608939218214146, 6.06761955766815337044080302134, 6.81307415233890380188367944981, 7.73412055932866845395637101310, 8.453694666267075156775904370822, 9.566792617391321665404064092960, 9.926661367629454239054133172646

Graph of the ZZ-function along the critical line