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Properties

Label 2-1785-1.1-c1-0-50
Degree 22
Conductor 17851785
Sign 11
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·2-s + 3-s + 4.56·4-s − 5-s + 2.56·6-s − 7-s + 6.56·8-s + 9-s − 2.56·10-s + 3.12·11-s + 4.56·12-s + 3.56·13-s − 2.56·14-s − 15-s + 7.68·16-s − 17-s + 2.56·18-s − 7.12·19-s − 4.56·20-s − 21-s + 8·22-s + 8.68·23-s + 6.56·24-s + 25-s + 9.12·26-s + 27-s − 4.56·28-s + ⋯
L(s)  = 1  + 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.447·5-s + 1.04·6-s − 0.377·7-s + 2.31·8-s + 0.333·9-s − 0.810·10-s + 0.941·11-s + 1.31·12-s + 0.987·13-s − 0.684·14-s − 0.258·15-s + 1.92·16-s − 0.242·17-s + 0.603·18-s − 1.63·19-s − 1.01·20-s − 0.218·21-s + 1.70·22-s + 1.81·23-s + 1.33·24-s + 0.200·25-s + 1.78·26-s + 0.192·27-s − 0.862·28-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), 1)(2,\ 1785,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.1784064536.178406453
L(12)L(\frac12) ≈\approx 6.1784064536.178406453
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)
bad3 1−T 1 - T
5 1+T 1 + T
7 1+T 1 + T
17 1+T 1 + T
good2 1−2.56T+2T2 1 - 2.56T + 2T^{2}
11 1−3.12T+11T2 1 - 3.12T + 11T^{2}
13 1−3.56T+13T2 1 - 3.56T + 13T^{2}
19 1+7.12T+19T2 1 + 7.12T + 19T^{2}
23 1−8.68T+23T2 1 - 8.68T + 23T^{2}
29 1+1.12T+29T2 1 + 1.12T + 29T^{2}
31 1−2.43T+31T2 1 - 2.43T + 31T^{2}
37 1+3.56T+37T2 1 + 3.56T + 37T^{2}
41 1−3.56T+41T2 1 - 3.56T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+11.8T+47T2 1 + 11.8T + 47T^{2}
53 1+2.87T+53T2 1 + 2.87T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1+10.6T+61T2 1 + 10.6T + 61T^{2}
67 1+6.24T+67T2 1 + 6.24T + 67T^{2}
71 1+7.12T+71T2 1 + 7.12T + 71T^{2}
73 1−2.87T+73T2 1 - 2.87T + 73T^{2}
79 1−3.12T+79T2 1 - 3.12T + 79T^{2}
83 1+15.8T+83T2 1 + 15.8T + 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1−6T+97T2 1 - 6T + 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

−9.080658977392302749807939672320, −8.499489019713316489828287440884, −7.33749952364122450869751170006, −6.59294025916800309965365867661, −6.16043648869638976910396516887, −4.92432828712169903519685640178, −4.19363900142589069621769757381, −3.54488608103195909899912214561, −2.79062541351769277558221454342, −1.55971063092187508187260055572, 1.55971063092187508187260055572, 2.79062541351769277558221454342, 3.54488608103195909899912214561, 4.19363900142589069621769757381, 4.92432828712169903519685640178, 6.16043648869638976910396516887, 6.59294025916800309965365867661, 7.33749952364122450869751170006, 8.499489019713316489828287440884, 9.080658977392302749807939672320

Graph of the ZZ-function along the critical line