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Properties

Label 2-4680-5.4-c1-0-32
Degree 22
Conductor 46804680
Sign 0.193−0.981i0.193 - 0.981i
Analytic cond. 37.369937.3699
Root an. cond. 6.113096.11309
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  + (−0.432 + 2.19i)5-s + 2.86·11-s − i·13-s + 5.52i·17-s − 3.52·19-s − 7.52i·23-s + (−4.62 − 1.89i)25-s + 6.77·29-s + 5.72·31-s + 3.72i·37-s + 10.1·41-s − 5.52i·43-s + 8.65i·47-s + 7·49-s + 6.77i·53-s + ⋯
L(s)  = 1  + (−0.193 + 0.981i)5-s + 0.863·11-s − 0.277i·13-s + 1.33i·17-s − 0.808·19-s − 1.56i·23-s + (−0.925 − 0.379i)25-s + 1.25·29-s + 1.02·31-s + 0.613i·37-s + 1.58·41-s − 0.842i·43-s + 1.26i·47-s + 49-s + 0.930i·53-s + ⋯

Functional equation

Λ(s)=(4680s/2 ΓC(s) L(s)=((0.193−0.981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4680s/2 ΓC(s+1/2) L(s)=((0.193−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46804680    =    23⋅32⋅5⋅132^{3} \cdot 3^{2} \cdot 5 \cdot 13
Sign: 0.193−0.981i0.193 - 0.981i
Analytic conductor: 37.369937.3699
Root analytic conductor: 6.113096.11309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4680(2809,⋅)\chi_{4680} (2809, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4680, ( :1/2), 0.193−0.981i)(2,\ 4680,\ (\ :1/2),\ 0.193 - 0.981i)

Particular Values

L(1)L(1) ≈\approx 1.8010618541.801061854
L(12)L(\frac12) ≈\approx 1.8010618541.801061854
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 1
5 1+(0.432−2.19i)T 1 + (0.432 - 2.19i)T
13 1+iT 1 + iT
good7 1−7T2 1 - 7T^{2}
11 1−2.86T+11T2 1 - 2.86T + 11T^{2}
17 1−5.52iT−17T2 1 - 5.52iT - 17T^{2}
19 1+3.52T+19T2 1 + 3.52T + 19T^{2}
23 1+7.52iT−23T2 1 + 7.52iT - 23T^{2}
29 1−6.77T+29T2 1 - 6.77T + 29T^{2}
31 1−5.72T+31T2 1 - 5.72T + 31T^{2}
37 1−3.72iT−37T2 1 - 3.72iT - 37T^{2}
41 1−10.1T+41T2 1 - 10.1T + 41T^{2}
43 1+5.52iT−43T2 1 + 5.52iT - 43T^{2}
47 1−8.65iT−47T2 1 - 8.65iT - 47T^{2}
53 1−6.77iT−53T2 1 - 6.77iT - 53T^{2}
59 1−0.593T+59T2 1 - 0.593T + 59T^{2}
61 1+5.25T+61T2 1 + 5.25T + 61T^{2}
67 1−10.5iT−67T2 1 - 10.5iT - 67T^{2}
71 1−2.38T+71T2 1 - 2.38T + 71T^{2}
73 1−5.45iT−73T2 1 - 5.45iT - 73T^{2}
79 1+2.47T+79T2 1 + 2.47T + 79T^{2}
83 1+8.11iT−83T2 1 + 8.11iT - 83T^{2}
89 1+14.1T+89T2 1 + 14.1T + 89T^{2}
97 1+6iT−97T2 1 + 6iT - 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

−8.425120743073341628868273339490, −7.79400609123136420726740042124, −6.83739152639280979666889543239, −6.35304221367034421979273785517, −5.85481012593241506975413612539, −4.38394205765901716660828495813, −4.13842918912213600175080120516, −2.97956226764813729100427102126, −2.33132382598715334318289327529, −1.03386278589994126815067191048, 0.59243593757709679059700618896, 1.54079912051774152703859307490, 2.65458032746447290560620701111, 3.75685407349228408381358467755, 4.43441508361631149398344838610, 5.09840545324127403708726290904, 5.93020585797601907393633195346, 6.72220424177298598955374694829, 7.47260936783346338712087983172, 8.200723662411147341530368792946

Graph of the ZZ-function along the critical line