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Properties

Label 2-1320-5.4-c1-0-31
Degree 22
Conductor 13201320
Sign −0.848−0.529i-0.848 - 0.529i
Analytic cond. 10.540210.5402
Root an. cond. 3.246573.24657
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  − i·3-s + (−1.89 − 1.18i)5-s − 4.20i·7-s − 9-s − 11-s − 2.90i·13-s + (−1.18 + 1.89i)15-s − 2.41i·17-s − 6.16·19-s − 4.20·21-s + 8.05i·23-s + (2.19 + 4.49i)25-s + i·27-s + 5.68·29-s + 1.52·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.848 − 0.529i)5-s − 1.58i·7-s − 0.333·9-s − 0.301·11-s − 0.804i·13-s + (−0.305 + 0.489i)15-s − 0.585i·17-s − 1.41·19-s − 0.917·21-s + 1.67i·23-s + (0.438 + 0.898i)25-s + 0.192i·27-s + 1.05·29-s + 0.273·31-s + ⋯

Functional equation

Λ(s)=(1320s/2 ΓC(s) L(s)=((−0.848−0.529i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1320s/2 ΓC(s+1/2) L(s)=((−0.848−0.529i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13201320    =    23⋅3⋅5⋅112^{3} \cdot 3 \cdot 5 \cdot 11
Sign: −0.848−0.529i-0.848 - 0.529i
Analytic conductor: 10.540210.5402
Root analytic conductor: 3.246573.24657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1320(529,⋅)\chi_{1320} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1320, ( :1/2), −0.848−0.529i)(2,\ 1320,\ (\ :1/2),\ -0.848 - 0.529i)

Particular Values

L(1)L(1) ≈\approx 0.59651831590.5965183159
L(12)L(\frac12) ≈\approx 0.59651831590.5965183159
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
3 1+iT 1 + iT
5 1+(1.89+1.18i)T 1 + (1.89 + 1.18i)T
11 1+T 1 + T
good7 1+4.20iT−7T2 1 + 4.20iT - 7T^{2}
13 1+2.90iT−13T2 1 + 2.90iT - 13T^{2}
17 1+2.41iT−17T2 1 + 2.41iT - 17T^{2}
19 1+6.16T+19T2 1 + 6.16T + 19T^{2}
23 1−8.05iT−23T2 1 - 8.05iT - 23T^{2}
29 1−5.68T+29T2 1 - 5.68T + 29T^{2}
31 1−1.52T+31T2 1 - 1.52T + 31T^{2}
37 1−0.577iT−37T2 1 - 0.577iT - 37T^{2}
41 1−3.30T+41T2 1 - 3.30T + 41T^{2}
43 1+10.6iT−43T2 1 + 10.6iT - 43T^{2}
47 1+0.355iT−47T2 1 + 0.355iT - 47T^{2}
53 1−9.95iT−53T2 1 - 9.95iT - 53T^{2}
59 1+10.8T+59T2 1 + 10.8T + 59T^{2}
61 1+8.90T+61T2 1 + 8.90T + 61T^{2}
67 1−5.40iT−67T2 1 - 5.40iT - 67T^{2}
71 1−6.61T+71T2 1 - 6.61T + 71T^{2}
73 1−6.21iT−73T2 1 - 6.21iT - 73T^{2}
79 1+13.4T+79T2 1 + 13.4T + 79T^{2}
83 1+10.7iT−83T2 1 + 10.7iT - 83T^{2}
89 1+7.52T+89T2 1 + 7.52T + 89T^{2}
97 1−3.33iT−97T2 1 - 3.33iT - 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

−8.983816211202246066379738589154, −8.135429522343422851480260021407, −7.50432389549228349771855314591, −7.02326788442823698069241389314, −5.84864328016698904943527096700, −4.74590772350914974716326111534, −3.99227897769908756969761962892, −3.00166266735070065746905819549, −1.32260103586234665145656891242, −0.25756743806928078009709982620, 2.24427361220912339731730182142, 2.98826002543314049682404789603, 4.25283259754301588656620124051, 4.83027951515077042349212764689, 6.21302965367055574824252607822, 6.50445271082852464531645814025, 7.992723719607109670433338616025, 8.509131147322587580971286097414, 9.144457357324155084261144341266, 10.20039873826528918230133508866

Graph of the ZZ-function along the critical line