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Properties

Label 2-336-7.2-c5-0-13
Degree 22
Conductor 336336
Sign 0.972−0.233i0.972 - 0.233i
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
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  + (−4.5 − 7.79i)3-s + (37.7 − 65.3i)5-s + (−99.4 + 83.1i)7-s + (−40.5 + 70.1i)9-s + (−74.7 − 129. i)11-s + 349.·13-s − 679.·15-s + (574. + 995. i)17-s + (−1.39e3 + 2.42e3i)19-s + (1.09e3 + 401. i)21-s + (906. − 1.57e3i)23-s + (−1.28e3 − 2.22e3i)25-s + 729·27-s − 759.·29-s + (4.51e3 + 7.82e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.675 − 1.16i)5-s + (−0.767 + 0.641i)7-s + (−0.166 + 0.288i)9-s + (−0.186 − 0.322i)11-s + 0.573·13-s − 0.779·15-s + (0.482 + 0.835i)17-s + (−0.888 + 1.53i)19-s + (0.542 + 0.198i)21-s + (0.357 − 0.619i)23-s + (−0.411 − 0.713i)25-s + 0.192·27-s − 0.167·29-s + (0.843 + 1.46i)31-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((0.972−0.233i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=((0.972−0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 0.972−0.233i0.972 - 0.233i
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ336(289,⋅)\chi_{336} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 336, ( :5/2), 0.972−0.233i)(2,\ 336,\ (\ :5/2),\ 0.972 - 0.233i)

Particular Values

L(3)L(3) ≈\approx 1.6263315841.626331584
L(12)L(\frac12) ≈\approx 1.6263315841.626331584
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
3 1+(4.5+7.79i)T 1 + (4.5 + 7.79i)T
7 1+(99.4−83.1i)T 1 + (99.4 - 83.1i)T
good5 1+(−37.7+65.3i)T+(−1.56e3−2.70e3i)T2 1 + (-37.7 + 65.3i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(74.7+129.i)T+(−8.05e4+1.39e5i)T2 1 + (74.7 + 129. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−349.T+3.71e5T2 1 - 349.T + 3.71e5T^{2}
17 1+(−574.−995.i)T+(−7.09e5+1.22e6i)T2 1 + (-574. - 995. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(1.39e3−2.42e3i)T+(−1.23e6−2.14e6i)T2 1 + (1.39e3 - 2.42e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−906.+1.57e3i)T+(−3.21e6−5.57e6i)T2 1 + (-906. + 1.57e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+759.T+2.05e7T2 1 + 759.T + 2.05e7T^{2}
31 1+(−4.51e3−7.82e3i)T+(−1.43e7+2.47e7i)T2 1 + (-4.51e3 - 7.82e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(3.89e3−6.75e3i)T+(−3.46e7−6.00e7i)T2 1 + (3.89e3 - 6.75e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−7.64e3T+1.15e8T2 1 - 7.64e3T + 1.15e8T^{2}
43 1+1.21e4T+1.47e8T2 1 + 1.21e4T + 1.47e8T^{2}
47 1+(−1.22e4+2.13e4i)T+(−1.14e8−1.98e8i)T2 1 + (-1.22e4 + 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(6.79e3+1.17e4i)T+(−2.09e8+3.62e8i)T2 1 + (6.79e3 + 1.17e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.31e4+2.28e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.31e4 + 2.28e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(1.76e4−3.05e4i)T+(−4.22e8−7.31e8i)T2 1 + (1.76e4 - 3.05e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−2.71e4−4.70e4i)T+(−6.75e8+1.16e9i)T2 1 + (-2.71e4 - 4.70e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−7.01e4T+1.80e9T2 1 - 7.01e4T + 1.80e9T^{2}
73 1+(−2.22e4−3.85e4i)T+(−1.03e9+1.79e9i)T2 1 + (-2.22e4 - 3.85e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−3.08e4+5.33e4i)T+(−1.53e9−2.66e9i)T2 1 + (-3.08e4 + 5.33e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−8.71e4T+3.93e9T2 1 - 8.71e4T + 3.93e9T^{2}
89 1+(4.92e4−8.53e4i)T+(−2.79e9−4.83e9i)T2 1 + (4.92e4 - 8.53e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−3.23e4T+8.58e9T2 1 - 3.23e4T + 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

−10.62218332178224995604072852903, −9.855684380858355099784442124078, −8.619437599632245049616289137901, −8.328901825825577667572612221965, −6.58710765598126271228871293941, −5.90946677805727893461049424890, −5.08141941541357539426494216769, −3.55264311809499037253193918691, −2.00646339001160904213629184629, −0.974320334314310470347957917689, 0.52552397027707958956902202247, 2.45068996262327408558974267907, 3.39954939959838367557642014922, 4.64945514098815285281246171749, 6.00202521184284680332592697544, 6.69144271671169524909719445782, 7.58876696990920585690303417496, 9.265114367739517182586482014100, 9.769250592106327629818084999368, 10.80725770303247811433520051120

Graph of the ZZ-function along the critical line