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Properties

Label 2-392-7.4-c5-0-45
Degree 22
Conductor 392392
Sign −0.701+0.712i-0.701 + 0.712i
Analytic cond. 62.870462.8704
Root an. cond. 7.929087.92908
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  + (15 − 25.9i)3-s + (16 + 27.7i)5-s + (−328.5 − 568. i)9-s + (312 − 540. i)11-s + 708·13-s + 960·15-s + (467 − 808. i)17-s + (929 + 1.60e3i)19-s + (560 + 969. i)23-s + (1.05e3 − 1.81e3i)25-s − 1.24e4·27-s − 1.17e3·29-s + (1.45e3 − 2.51e3i)31-s + (−9.36e3 − 1.62e4i)33-s + (6.23e3 + 1.07e4i)37-s + ⋯
L(s)  = 1  + (0.962 − 1.66i)3-s + (0.286 + 0.495i)5-s + (−1.35 − 2.34i)9-s + (0.777 − 1.34i)11-s + 1.16·13-s + 1.10·15-s + (0.391 − 0.678i)17-s + (0.590 + 1.02i)19-s + (0.220 + 0.382i)23-s + (0.336 − 0.582i)25-s − 3.27·27-s − 0.259·29-s + (0.271 − 0.470i)31-s + (−1.49 − 2.59i)33-s + (0.748 + 1.29i)37-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((−0.701+0.712i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+5/2) L(s)=((−0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: −0.701+0.712i-0.701 + 0.712i
Analytic conductor: 62.870462.8704
Root analytic conductor: 7.929087.92908
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ392(361,⋅)\chi_{392} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 392, ( :5/2), −0.701+0.712i)(2,\ 392,\ (\ :5/2),\ -0.701 + 0.712i)

Particular Values

L(3)L(3) ≈\approx 3.4717437283.471743728
L(12)L(\frac12) ≈\approx 3.4717437283.471743728
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
7 1 1
good3 1+(−15+25.9i)T+(−121.5−210.i)T2 1 + (-15 + 25.9i)T + (-121.5 - 210. i)T^{2}
5 1+(−16−27.7i)T+(−1.56e3+2.70e3i)T2 1 + (-16 - 27.7i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−312+540.i)T+(−8.05e4−1.39e5i)T2 1 + (-312 + 540. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−708T+3.71e5T2 1 - 708T + 3.71e5T^{2}
17 1+(−467+808.i)T+(−7.09e5−1.22e6i)T2 1 + (-467 + 808. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−929−1.60e3i)T+(−1.23e6+2.14e6i)T2 1 + (-929 - 1.60e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−560−969.i)T+(−3.21e6+5.57e6i)T2 1 + (-560 - 969. i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+1.17e3T+2.05e7T2 1 + 1.17e3T + 2.05e7T^{2}
31 1+(−1.45e3+2.51e3i)T+(−1.43e7−2.47e7i)T2 1 + (-1.45e3 + 2.51e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−6.23e3−1.07e4i)T+(−3.46e7+6.00e7i)T2 1 + (-6.23e3 - 1.07e4i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+2.66e3T+1.15e8T2 1 + 2.66e3T + 1.15e8T^{2}
43 1+7.14e3T+1.47e8T2 1 + 7.14e3T + 1.47e8T^{2}
47 1+(3.73e3+6.46e3i)T+(−1.14e8+1.98e8i)T2 1 + (3.73e3 + 6.46e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.36e4+2.36e4i)T+(−2.09e8−3.62e8i)T2 1 + (-1.36e4 + 2.36e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−1.24e3+2.15e3i)T+(−3.57e8−6.19e8i)T2 1 + (-1.24e3 + 2.15e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(5.54e3+9.60e3i)T+(−4.22e8+7.31e8i)T2 1 + (5.54e3 + 9.60e3i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(1.98e4−3.44e4i)T+(−6.75e8−1.16e9i)T2 1 + (1.98e4 - 3.44e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+6.98e4T+1.80e9T2 1 + 6.98e4T + 1.80e9T^{2}
73 1+(−8.22e3+1.42e4i)T+(−1.03e9−1.79e9i)T2 1 + (-8.22e3 + 1.42e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(3.91e4+6.78e4i)T+(−1.53e9+2.66e9i)T2 1 + (3.91e4 + 6.78e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+1.09e5T+3.93e9T2 1 + 1.09e5T + 3.93e9T^{2}
89 1+(2.84e4+4.93e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.84e4 + 4.93e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.15e5T+8.58e9T2 1 - 1.15e5T + 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

−9.990966490986119600583242081318, −8.837458445065026202240993100383, −8.309740891849449803749390683242, −7.36741254285767144816888849207, −6.38705977097259824679989232719, −5.89652998145406256732984954848, −3.55327563028368203396209140894, −2.95253941380448723378189919022, −1.56768343715656887228263965510, −0.792491763590318529497716301782, 1.54309185522367095235590273354, 2.92967951215178428732109902065, 4.01136531546143074472315228788, 4.67713129464723473405988880085, 5.72489597203325049824351645442, 7.28605677803584097834178864814, 8.529984197786673124661277343918, 9.099843257146321143380638974532, 9.727390410789578591845262431251, 10.59094767246149822484017994077

Graph of the ZZ-function along the critical line