↑

Properties

Label 2-538-269.82-c2-0-40
Degree 22
Conductor 538538
Sign −0.466+0.884i-0.466 + 0.884i
Analytic cond. 14.659414.6594
Root an. cond. 3.828763.82876
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (1.93 − 1.93i)3-s − 2i·4-s + 9.46·5-s − 3.87i·6-s + (−8.10 − 8.10i)7-s + (−2 − 2i)8-s + 1.51i·9-s + (9.46 − 9.46i)10-s − 1.78i·11-s + (−3.87 − 3.87i)12-s − 5.85i·13-s − 16.2·14-s + (18.3 − 18.3i)15-s − 4·16-s + (−3.08 + 3.08i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.645 − 0.645i)3-s − 0.5i·4-s + 1.89·5-s − 0.645i·6-s + (−1.15 − 1.15i)7-s + (−0.250 − 0.250i)8-s + 0.167i·9-s + (0.946 − 0.946i)10-s − 0.162i·11-s + (−0.322 − 0.322i)12-s − 0.450i·13-s − 1.15·14-s + (1.22 − 1.22i)15-s − 0.250·16-s + (−0.181 + 0.181i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.466+0.884i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1) L(s)=((−0.466+0.884i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.466+0.884i-0.466 + 0.884i
Analytic conductor: 14.659414.6594
Root analytic conductor: 3.828763.82876
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ538(351,⋅)\chi_{538} (351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1), −0.466+0.884i)(2,\ 538,\ (\ :1),\ -0.466 + 0.884i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.3875729093.387572909
L(12)L(\frac12) ≈\approx 3.3875729093.387572909
L(2)L(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+i)T 1 + (-1 + i)T
269 1+(−157.−217.i)T 1 + (-157. - 217. i)T
good3 1+(−1.93+1.93i)T−9iT2 1 + (-1.93 + 1.93i)T - 9iT^{2}
5 1−9.46T+25T2 1 - 9.46T + 25T^{2}
7 1+(8.10+8.10i)T+49iT2 1 + (8.10 + 8.10i)T + 49iT^{2}
11 1+1.78iT−121T2 1 + 1.78iT - 121T^{2}
13 1+5.85iT−169T2 1 + 5.85iT - 169T^{2}
17 1+(3.08−3.08i)T−289iT2 1 + (3.08 - 3.08i)T - 289iT^{2}
19 1+(13.1+13.1i)T+361iT2 1 + (13.1 + 13.1i)T + 361iT^{2}
23 1+2.18T+529T2 1 + 2.18T + 529T^{2}
29 1+(1.77+1.77i)T+841iT2 1 + (1.77 + 1.77i)T + 841iT^{2}
31 1+(−5.27−5.27i)T+961iT2 1 + (-5.27 - 5.27i)T + 961iT^{2}
37 1−11.7T+1.36e3T2 1 - 11.7T + 1.36e3T^{2}
41 1−38.1T+1.68e3T2 1 - 38.1T + 1.68e3T^{2}
43 1−61.6iT−1.84e3T2 1 - 61.6iT - 1.84e3T^{2}
47 1−65.9T+2.20e3T2 1 - 65.9T + 2.20e3T^{2}
53 1−82.5T+2.80e3T2 1 - 82.5T + 2.80e3T^{2}
59 1+(49.0+49.0i)T+3.48e3iT2 1 + (49.0 + 49.0i)T + 3.48e3iT^{2}
61 1+97.6T+3.72e3T2 1 + 97.6T + 3.72e3T^{2}
67 1+33.0T+4.48e3T2 1 + 33.0T + 4.48e3T^{2}
71 1+(−18.4−18.4i)T+5.04e3iT2 1 + (-18.4 - 18.4i)T + 5.04e3iT^{2}
73 1−104.iT−5.32e3T2 1 - 104. iT - 5.32e3T^{2}
79 1+138.iT−6.24e3T2 1 + 138. iT - 6.24e3T^{2}
83 1+(106.+106.i)T+6.88e3iT2 1 + (106. + 106. i)T + 6.88e3iT^{2}
89 1−42.1iT−7.92e3T2 1 - 42.1iT - 7.92e3T^{2}
97 1−100.iT−9.40e3T2 1 - 100. iT - 9.40e3T^{2}
show more
show less
   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

−10.39523494378001859674289113302, −9.590794044502627811734011664673, −8.828867778965350308167185590342, −7.40209503407748429509490275753, −6.51347833284497536234272592415, −5.84568758280971946466050486191, −4.54173775205652796755190687255, −3.06436578937333718417239766242, −2.30827360156037364742093200806, −1.06483716049526429949223088122, 2.20183305945562029290432453938, 2.95669936021456496093109886469, 4.26538232615527459021314671958, 5.66232722200713744442415811442, 6.04708995395022883104058874577, 6.92372720470021171268325220680, 8.693585160311451596000748609801, 9.142107024395618751099674809220, 9.742583247087184806664439387649, 10.52640504377591832909349485179

Graph of the ZZ-function along the critical line