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Properties

Label 2-84-7.2-c5-0-3
Degree 22
Conductor 8484
Sign 0.706−0.707i0.706 - 0.707i
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
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 + (15.9 − 27.6i)5-s + (85.7 + 97.2i)7-s + (−40.5 + 70.1i)9-s + (−130. − 225. i)11-s + 769.·13-s + 287.·15-s + (776. + 1.34e3i)17-s + (−375. + 649. i)19-s + (−372. + 1.10e3i)21-s + (−377. + 653. i)23-s + (1.05e3 + 1.82e3i)25-s − 729·27-s + 6.00e3·29-s + (3.21e3 + 5.55e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.285 − 0.495i)5-s + (0.661 + 0.750i)7-s + (−0.166 + 0.288i)9-s + (−0.325 − 0.562i)11-s + 1.26·13-s + 0.330·15-s + (0.651 + 1.12i)17-s + (−0.238 + 0.412i)19-s + (−0.184 + 0.547i)21-s + (−0.148 + 0.257i)23-s + (0.336 + 0.582i)25-s − 0.192·27-s + 1.32·29-s + (0.599 + 1.03i)31-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.706−0.707i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((0.706−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.706−0.707i0.706 - 0.707i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(37,⋅)\chi_{84} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), 0.706−0.707i)(2,\ 84,\ (\ :5/2),\ 0.706 - 0.707i)

Particular Values

L(3)L(3) ≈\approx 2.08237+0.862844i2.08237 + 0.862844i
L(12)L(\frac12) ≈\approx 2.08237+0.862844i2.08237 + 0.862844i
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+(−85.7−97.2i)T 1 + (-85.7 - 97.2i)T
good5 1+(−15.9+27.6i)T+(−1.56e3−2.70e3i)T2 1 + (-15.9 + 27.6i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(130.+225.i)T+(−8.05e4+1.39e5i)T2 1 + (130. + 225. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−769.T+3.71e5T2 1 - 769.T + 3.71e5T^{2}
17 1+(−776.−1.34e3i)T+(−7.09e5+1.22e6i)T2 1 + (-776. - 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(375.−649.i)T+(−1.23e6−2.14e6i)T2 1 + (375. - 649. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(377.−653.i)T+(−3.21e6−5.57e6i)T2 1 + (377. - 653. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−6.00e3T+2.05e7T2 1 - 6.00e3T + 2.05e7T^{2}
31 1+(−3.21e3−5.55e3i)T+(−1.43e7+2.47e7i)T2 1 + (-3.21e3 - 5.55e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−2.38e3+4.13e3i)T+(−3.46e7−6.00e7i)T2 1 + (-2.38e3 + 4.13e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+5.42e3T+1.15e8T2 1 + 5.42e3T + 1.15e8T^{2}
43 1+1.18e4T+1.47e8T2 1 + 1.18e4T + 1.47e8T^{2}
47 1+(−8.71e3+1.50e4i)T+(−1.14e8−1.98e8i)T2 1 + (-8.71e3 + 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(1.88e4+3.26e4i)T+(−2.09e8+3.62e8i)T2 1 + (1.88e4 + 3.26e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.10e4+1.91e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.10e4 + 1.91e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−4.08e3+7.07e3i)T+(−4.22e8−7.31e8i)T2 1 + (-4.08e3 + 7.07e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(6.50e3+1.12e4i)T+(−6.75e8+1.16e9i)T2 1 + (6.50e3 + 1.12e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+1.23e4T+1.80e9T2 1 + 1.23e4T + 1.80e9T^{2}
73 1+(2.18e4+3.77e4i)T+(−1.03e9+1.79e9i)T2 1 + (2.18e4 + 3.77e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(3.83e4−6.64e4i)T+(−1.53e9−2.66e9i)T2 1 + (3.83e4 - 6.64e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−2.18e4T+3.93e9T2 1 - 2.18e4T + 3.93e9T^{2}
89 1+(−6.84e4+1.18e5i)T+(−2.79e9−4.83e9i)T2 1 + (-6.84e4 + 1.18e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+9.30e4T+8.58e9T2 1 + 9.30e4T + 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

−13.51605742690924285867595647623, −12.38804560447090398750272309344, −11.16912294207430047356182819325, −10.14607842283553332231022722423, −8.675183927893179770482470021399, −8.269299887435917048780181716560, −6.10208303215763370207627473459, −5.03404498602410703001881730994, −3.41957957351804240929443631520, −1.53819516409194577005221787439, 1.08328569138545676589532022316, 2.77977641546340569007566674488, 4.54282996525474825334978965668, 6.29207697722968644524151047594, 7.40914074779058748453269482407, 8.450270798764431032088715076058, 9.956867790221952783321740832168, 10.96174922226702206192855065815, 12.08277487345164215345836449691, 13.49053382475129142357458678149

Graph of the ZZ-function along the critical line