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Properties

Label 2-1785-5.4-c1-0-54
Degree 22
Conductor 17851785
Sign 0.388−0.921i0.388 - 0.921i
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  + 1.61i·2-s − i·3-s − 0.608·4-s + (2.06 + 0.867i)5-s + 1.61·6-s − i·7-s + 2.24i·8-s − 9-s + (−1.40 + 3.32i)10-s + 3.70·11-s + 0.608i·12-s − 0.484i·13-s + 1.61·14-s + (0.867 − 2.06i)15-s − 4.84·16-s − i·17-s + ⋯
L(s)  = 1  + 1.14i·2-s − 0.577i·3-s − 0.304·4-s + (0.921 + 0.388i)5-s + 0.659·6-s − 0.377i·7-s + 0.794i·8-s − 0.333·9-s + (−0.443 + 1.05i)10-s + 1.11·11-s + 0.175i·12-s − 0.134i·13-s + 0.431·14-s + (0.224 − 0.532i)15-s − 1.21·16-s − 0.242i·17-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=((0.388−0.921i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=((0.388−0.921i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: 0.388−0.921i0.388 - 0.921i
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), 0.388−0.921i)(2,\ 1785,\ (\ :1/2),\ 0.388 - 0.921i)

Particular Values

L(1)L(1) ≈\approx 2.4710271882.471027188
L(12)L(\frac12) ≈\approx 2.4710271882.471027188
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+(−2.06−0.867i)T 1 + (-2.06 - 0.867i)T
7 1+iT 1 + iT
17 1+iT 1 + iT
good2 1−1.61iT−2T2 1 - 1.61iT - 2T^{2}
11 1−3.70T+11T2 1 - 3.70T + 11T^{2}
13 1+0.484iT−13T2 1 + 0.484iT - 13T^{2}
19 1−8.59T+19T2 1 - 8.59T + 19T^{2}
23 1+6.32iT−23T2 1 + 6.32iT - 23T^{2}
29 1−5.03T+29T2 1 - 5.03T + 29T^{2}
31 1+10.3T+31T2 1 + 10.3T + 31T^{2}
37 1−5.74iT−37T2 1 - 5.74iT - 37T^{2}
41 1+4.52T+41T2 1 + 4.52T + 41T^{2}
43 1+0.961iT−43T2 1 + 0.961iT - 43T^{2}
47 1+3.97iT−47T2 1 + 3.97iT - 47T^{2}
53 1+3.25iT−53T2 1 + 3.25iT - 53T^{2}
59 1+0.477T+59T2 1 + 0.477T + 59T^{2}
61 1−1.83T+61T2 1 - 1.83T + 61T^{2}
67 1−3.45iT−67T2 1 - 3.45iT - 67T^{2}
71 1−8.74T+71T2 1 - 8.74T + 71T^{2}
73 1−10.6iT−73T2 1 - 10.6iT - 73T^{2}
79 1−16.1T+79T2 1 - 16.1T + 79T^{2}
83 1−5.53iT−83T2 1 - 5.53iT - 83T^{2}
89 1−4.43T+89T2 1 - 4.43T + 89T^{2}
97 1+12.0iT−97T2 1 + 12.0iT - 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

−9.241567975147860344187497011455, −8.482355936759903620827611752975, −7.58834627300548962176397076121, −6.79283096110699031051814618019, −6.60136425738142760693986500659, −5.58665893637913569100012199845, −4.98183978302497058301683071610, −3.48655472222029071924114872499, −2.38420246397756472316126905504, −1.21866077949614307075561432787, 1.16254302163613460598348687825, 1.97812764802280684224778411588, 3.17360269074546208372165286336, 3.80664602886901444203971388914, 4.97843830843460672473728694035, 5.71370621763195204526570876843, 6.61486243166397282514942285066, 7.60030935868463470088570142019, 8.973510002434140901970739654106, 9.384787482571778517227223591692

Graph of the ZZ-function along the critical line