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Properties

Label 2-13e2-13.12-c7-0-12
Degree 22
Conductor 169169
Sign −0.832+0.554i-0.832 + 0.554i
Analytic cond. 52.793052.7930
Root an. cond. 7.265887.26588
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 21.7i·2-s − 67.8·3-s − 346.·4-s + 157. i·5-s − 1.47e3i·6-s − 842. i·7-s − 4.77e3i·8-s + 2.41e3·9-s − 3.43e3·10-s + 1.99e3i·11-s + 2.35e4·12-s + 1.83e4·14-s − 1.07e4i·15-s + 5.96e4·16-s − 9.62e3·17-s + 5.26e4i·18-s + ⋯
L(s)  = 1  + 1.92i·2-s − 1.45·3-s − 2.71·4-s + 0.564i·5-s − 2.79i·6-s − 0.927i·7-s − 3.29i·8-s + 1.10·9-s − 1.08·10-s + 0.451i·11-s + 3.93·12-s + 1.78·14-s − 0.818i·15-s + 3.63·16-s − 0.474·17-s + 2.12i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.832+0.554i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((−0.832+0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.832+0.554i-0.832 + 0.554i
Analytic conductor: 52.793052.7930
Root analytic conductor: 7.265887.26588
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :7/2), −0.832+0.554i)(2,\ 169,\ (\ :7/2),\ -0.832 + 0.554i)

Particular Values

L(4)L(4) ≈\approx 0.50633897390.5063389739
L(12)L(\frac12) ≈\approx 0.50633897390.5063389739
L(92)L(\frac{9}{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)
bad13 1 1
good2 1−21.7iT−128T2 1 - 21.7iT - 128T^{2}
3 1+67.8T+2.18e3T2 1 + 67.8T + 2.18e3T^{2}
5 1−157.iT−7.81e4T2 1 - 157. iT - 7.81e4T^{2}
7 1+842.iT−8.23e5T2 1 + 842. iT - 8.23e5T^{2}
11 1−1.99e3iT−1.94e7T2 1 - 1.99e3iT - 1.94e7T^{2}
17 1+9.62e3T+4.10e8T2 1 + 9.62e3T + 4.10e8T^{2}
19 1+8.43e3iT−8.93e8T2 1 + 8.43e3iT - 8.93e8T^{2}
23 1+3.41e4T+3.40e9T2 1 + 3.41e4T + 3.40e9T^{2}
29 1+9.67e3T+1.72e10T2 1 + 9.67e3T + 1.72e10T^{2}
31 1−2.05e5iT−2.75e10T2 1 - 2.05e5iT - 2.75e10T^{2}
37 1+4.32e5iT−9.49e10T2 1 + 4.32e5iT - 9.49e10T^{2}
41 1−1.13e5iT−1.94e11T2 1 - 1.13e5iT - 1.94e11T^{2}
43 1−4.00e5T+2.71e11T2 1 - 4.00e5T + 2.71e11T^{2}
47 1+1.32e6iT−5.06e11T2 1 + 1.32e6iT - 5.06e11T^{2}
53 1−2.98e5T+1.17e12T2 1 - 2.98e5T + 1.17e12T^{2}
59 1−1.63e6iT−2.48e12T2 1 - 1.63e6iT - 2.48e12T^{2}
61 1−1.66e6T+3.14e12T2 1 - 1.66e6T + 3.14e12T^{2}
67 1−1.42e6iT−6.06e12T2 1 - 1.42e6iT - 6.06e12T^{2}
71 1+1.03e6iT−9.09e12T2 1 + 1.03e6iT - 9.09e12T^{2}
73 1+4.43e6iT−1.10e13T2 1 + 4.43e6iT - 1.10e13T^{2}
79 1+7.40e6T+1.92e13T2 1 + 7.40e6T + 1.92e13T^{2}
83 1−7.53e5iT−2.71e13T2 1 - 7.53e5iT - 2.71e13T^{2}
89 1−2.49e6iT−4.42e13T2 1 - 2.49e6iT - 4.42e13T^{2}
97 1−6.38e6iT−8.07e13T2 1 - 6.38e6iT - 8.07e13T^{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

−12.26592280105850207415038410538, −10.82013289829489040070686441068, −10.06943350108820541086616860566, −8.723560789737594595956306584822, −7.25411319092295426565125530199, −6.91807077785162149522162078214, −5.92506626883978693998823645193, −4.96894516058478244759720637116, −4.01903669750111846183870833477, −0.71934860822066789842772638971, 0.28921127799695434504499636790, 1.30073364634274132939900258404, 2.63889412385859513178685277633, 4.21512762021340405202261159426, 5.15953638643277826779806251783, 6.01981061881754087743380353000, 8.321474831841921171922904510742, 9.262448285446477422502427477381, 10.23966474893366742941612405084, 11.22486633144736477578055809827

Graph of the ZZ-function along the critical line