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Properties

Label 2-1785-5.4-c1-0-94
Degree 22
Conductor 17851785
Sign −0.987−0.158i-0.987 - 0.158i
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
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  − 0.304i·2-s − i·3-s + 1.90·4-s + (0.355 − 2.20i)5-s − 0.304·6-s − i·7-s − 1.18i·8-s − 9-s + (−0.671 − 0.108i)10-s − 6.38·11-s − 1.90i·12-s + 1.12i·13-s − 0.304·14-s + (−2.20 − 0.355i)15-s + 3.45·16-s − i·17-s + ⋯
L(s)  = 1  − 0.215i·2-s − 0.577i·3-s + 0.953·4-s + (0.158 − 0.987i)5-s − 0.124·6-s − 0.377i·7-s − 0.420i·8-s − 0.333·9-s + (−0.212 − 0.0341i)10-s − 1.92·11-s − 0.550i·12-s + 0.312i·13-s − 0.0813·14-s + (−0.570 − 0.0916i)15-s + 0.863·16-s − 0.242i·17-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=((−0.987−0.158i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=((−0.987−0.158i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: −0.987−0.158i-0.987 - 0.158i
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1785(1429,⋅)\chi_{1785} (1429, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), −0.987−0.158i)(2,\ 1785,\ (\ :1/2),\ -0.987 - 0.158i)

Particular Values

L(1)L(1) ≈\approx 1.1671483251.167148325
L(12)L(\frac12) ≈\approx 1.1671483251.167148325
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+iT 1 + iT
5 1+(−0.355+2.20i)T 1 + (-0.355 + 2.20i)T
7 1+iT 1 + iT
17 1+iT 1 + iT
good2 1+0.304iT−2T2 1 + 0.304iT - 2T^{2}
11 1+6.38T+11T2 1 + 6.38T + 11T^{2}
13 1−1.12iT−13T2 1 - 1.12iT - 13T^{2}
19 1+5.66T+19T2 1 + 5.66T + 19T^{2}
23 1−0.581iT−23T2 1 - 0.581iT - 23T^{2}
29 1+3.76T+29T2 1 + 3.76T + 29T^{2}
31 1−0.646T+31T2 1 - 0.646T + 31T^{2}
37 1−5.95iT−37T2 1 - 5.95iT - 37T^{2}
41 1+5.31T+41T2 1 + 5.31T + 41T^{2}
43 1+4.23iT−43T2 1 + 4.23iT - 43T^{2}
47 1+5.25iT−47T2 1 + 5.25iT - 47T^{2}
53 1+13.3iT−53T2 1 + 13.3iT - 53T^{2}
59 1−6.85T+59T2 1 - 6.85T + 59T^{2}
61 1−14.0T+61T2 1 - 14.0T + 61T^{2}
67 1−6.03iT−67T2 1 - 6.03iT - 67T^{2}
71 1+8.66T+71T2 1 + 8.66T + 71T^{2}
73 1+9.19iT−73T2 1 + 9.19iT - 73T^{2}
79 1−4.17T+79T2 1 - 4.17T + 79T^{2}
83 1−10.0iT−83T2 1 - 10.0iT - 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1+12.5iT−97T2 1 + 12.5iT - 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

−8.578860508093042786805374739607, −8.123382729593521244407586916672, −7.30023119287730498547507069564, −6.57681972569510620767441192572, −5.59339763725375980496336983943, −4.94329710165077794367201898805, −3.68446236209746252351084467474, −2.46088929123106853936319877471, −1.80018175227065106329118150079, −0.36939585687380614947259679685, 2.22409793754747032703751941227, 2.66879751712040475394040009070, 3.66848922362252952440260508693, 5.01851727651139635196186940347, 5.79923468061988693514387421919, 6.37901343247972712409562429276, 7.43854732138004211014357983294, 7.918360997139930132502077524765, 8.825025481993050207529217948326, 10.04259530047981185235677149328

Graph of the ZZ-function along the critical line