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Properties

Label 2-10e2-5.4-c5-0-3
Degree 22
Conductor 100100
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 16.038316.0383
Root an. cond. 4.004794.00479
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12i·3-s − 88i·7-s + 99·9-s + 540·11-s + 418i·13-s + 594i·17-s − 836·19-s + 1.05e3·21-s + 4.10e3i·23-s + 4.10e3i·27-s + 594·29-s + 4.25e3·31-s + 6.48e3i·33-s − 298i·37-s − 5.01e3·39-s + ⋯
L(s)  = 1  + 0.769i·3-s − 0.678i·7-s + 0.407·9-s + 1.34·11-s + 0.685i·13-s + 0.498i·17-s − 0.531·19-s + 0.522·21-s + 1.61i·23-s + 1.08i·27-s + 0.131·29-s + 0.795·31-s + 1.03i·33-s − 0.0357i·37-s − 0.528·39-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+5/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 16.038316.0383
Root analytic conductor: 4.004794.00479
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ100(49,⋅)\chi_{100} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 100, ( :5/2), 0.447−0.894i)(2,\ 100,\ (\ :5/2),\ 0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 1.68475+1.04123i1.68475 + 1.04123i
L(12)L(\frac12) ≈\approx 1.68475+1.04123i1.68475 + 1.04123i
L(72)L(\frac{7}{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
5 1 1
good3 1−12iT−243T2 1 - 12iT - 243T^{2}
7 1+88iT−1.68e4T2 1 + 88iT - 1.68e4T^{2}
11 1−540T+1.61e5T2 1 - 540T + 1.61e5T^{2}
13 1−418iT−3.71e5T2 1 - 418iT - 3.71e5T^{2}
17 1−594iT−1.41e6T2 1 - 594iT - 1.41e6T^{2}
19 1+836T+2.47e6T2 1 + 836T + 2.47e6T^{2}
23 1−4.10e3iT−6.43e6T2 1 - 4.10e3iT - 6.43e6T^{2}
29 1−594T+2.05e7T2 1 - 594T + 2.05e7T^{2}
31 1−4.25e3T+2.86e7T2 1 - 4.25e3T + 2.86e7T^{2}
37 1+298iT−6.93e7T2 1 + 298iT - 6.93e7T^{2}
41 1−1.72e4T+1.15e8T2 1 - 1.72e4T + 1.15e8T^{2}
43 1−1.21e4iT−1.47e8T2 1 - 1.21e4iT - 1.47e8T^{2}
47 1+1.29e3iT−2.29e8T2 1 + 1.29e3iT - 2.29e8T^{2}
53 1+1.94e4iT−4.18e8T2 1 + 1.94e4iT - 4.18e8T^{2}
59 1−7.66e3T+7.14e8T2 1 - 7.66e3T + 7.14e8T^{2}
61 1+3.47e4T+8.44e8T2 1 + 3.47e4T + 8.44e8T^{2}
67 1−2.18e4iT−1.35e9T2 1 - 2.18e4iT - 1.35e9T^{2}
71 1+4.68e4T+1.80e9T2 1 + 4.68e4T + 1.80e9T^{2}
73 1+6.75e4iT−2.07e9T2 1 + 6.75e4iT - 2.07e9T^{2}
79 1−7.69e4T+3.07e9T2 1 - 7.69e4T + 3.07e9T^{2}
83 1+6.77e4iT−3.93e9T2 1 + 6.77e4iT - 3.93e9T^{2}
89 1+2.97e4T+5.58e9T2 1 + 2.97e4T + 5.58e9T^{2}
97 1+1.22e5iT−8.58e9T2 1 + 1.22e5iT - 8.58e9T^{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

−13.20815479895680098001564353237, −11.89661695338397288241979451445, −10.89708806219967913385458770068, −9.822502134501987913785763487523, −9.026152304595149776645052717057, −7.44555903297766330524587557750, −6.27092091519014015084841797241, −4.50685572201736864303206024550, −3.71666978379047021140590370427, −1.40565260639191609453305475098, 0.913882871596800214533784300681, 2.46587976791816301424791845958, 4.33341920730405106686174517848, 6.03975676589006917803936088875, 6.97346389887750239379591014397, 8.274844268100986123986863333172, 9.335093011529123433969775117690, 10.65898617984728040008810232315, 12.05755404258012186985969404174, 12.51396853960862174236686145094

Graph of the ZZ-function along the critical line