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Properties

Label 2-55-5.4-c3-0-10
Degree 22
Conductor 5555
Sign −0.196+0.980i-0.196 + 0.980i
Analytic cond. 3.245103.24510
Root an. cond. 1.801411.80141
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.54i·2-s + 4.97i·3-s − 4.53·4-s + (2.19 − 10.9i)5-s + 17.6·6-s − 25.4i·7-s − 12.2i·8-s + 2.25·9-s + (−38.8 − 7.78i)10-s + 11·11-s − 22.5i·12-s + 80.0i·13-s − 90.0·14-s + (54.5 + 10.9i)15-s − 79.7·16-s − 59.1i·17-s + ⋯
L(s)  = 1  − 1.25i·2-s + 0.957i·3-s − 0.567·4-s + (0.196 − 0.980i)5-s + 1.19·6-s − 1.37i·7-s − 0.541i·8-s + 0.0836·9-s + (−1.22 − 0.246i)10-s + 0.301·11-s − 0.543i·12-s + 1.70i·13-s − 1.71·14-s + (0.938 + 0.188i)15-s − 1.24·16-s − 0.844i·17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((−0.196+0.980i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+3/2) L(s)=((−0.196+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: −0.196+0.980i-0.196 + 0.980i
Analytic conductor: 3.245103.24510
Root analytic conductor: 1.801411.80141
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ55(34,⋅)\chi_{55} (34, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :3/2), −0.196+0.980i)(2,\ 55,\ (\ :3/2),\ -0.196 + 0.980i)

Particular Values

L(2)L(2) ≈\approx 0.944251−1.15237i0.944251 - 1.15237i
L(12)L(\frac12) ≈\approx 0.944251−1.15237i0.944251 - 1.15237i
L(52)L(\frac{5}{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)
bad5 1+(−2.19+10.9i)T 1 + (-2.19 + 10.9i)T
11 1−11T 1 - 11T
good2 1+3.54iT−8T2 1 + 3.54iT - 8T^{2}
3 1−4.97iT−27T2 1 - 4.97iT - 27T^{2}
7 1+25.4iT−343T2 1 + 25.4iT - 343T^{2}
13 1−80.0iT−2.19e3T2 1 - 80.0iT - 2.19e3T^{2}
17 1+59.1iT−4.91e3T2 1 + 59.1iT - 4.91e3T^{2}
19 1−49.0T+6.85e3T2 1 - 49.0T + 6.85e3T^{2}
23 1−147.iT−1.21e4T2 1 - 147. iT - 1.21e4T^{2}
29 1−194.T+2.43e4T2 1 - 194.T + 2.43e4T^{2}
31 1+162.T+2.97e4T2 1 + 162.T + 2.97e4T^{2}
37 1−14.5iT−5.06e4T2 1 - 14.5iT - 5.06e4T^{2}
41 1−213.T+6.89e4T2 1 - 213.T + 6.89e4T^{2}
43 1−57.9iT−7.95e4T2 1 - 57.9iT - 7.95e4T^{2}
47 1−415.iT−1.03e5T2 1 - 415. iT - 1.03e5T^{2}
53 1+453.iT−1.48e5T2 1 + 453. iT - 1.48e5T^{2}
59 1−300.T+2.05e5T2 1 - 300.T + 2.05e5T^{2}
61 1−164.T+2.26e5T2 1 - 164.T + 2.26e5T^{2}
67 1−44.6iT−3.00e5T2 1 - 44.6iT - 3.00e5T^{2}
71 1+553.T+3.57e5T2 1 + 553.T + 3.57e5T^{2}
73 1+589.iT−3.89e5T2 1 + 589. iT - 3.89e5T^{2}
79 1+356.T+4.93e5T2 1 + 356.T + 4.93e5T^{2}
83 1−1.13e3iT−5.71e5T2 1 - 1.13e3iT - 5.71e5T^{2}
89 1−963.T+7.04e5T2 1 - 963.T + 7.04e5T^{2}
97 1−172.iT−9.12e5T2 1 - 172. iT - 9.12e5T^{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

−14.10420166792883920709627908144, −13.27137685354397942329642879818, −11.91863963219621564106021756469, −11.01918024076974446322310146670, −9.760232003893692089045603103479, −9.321567604342897045589731961106, −7.10813584121527462013932960447, −4.64825197862189326414311462939, −3.82126084346972839481196400130, −1.30286162813950110678075769210, 2.51866876697626996885549066096, 5.63128293474074103981010727275, 6.44520629588537444998094133134, 7.58308121817611805411730405022, 8.586221766501351742048715103306, 10.43851077647007655091995682209, 11.97335522448937323731992315446, 13.01695306667889321629003822196, 14.38363767077700723136407811543, 15.12408045348956245948849067696

Graph of the ZZ-function along the critical line