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Properties

Label 2-84-7.2-c5-0-4
Degree 22
Conductor 8484
Sign −0.580+0.814i-0.580 + 0.814i
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 + (−39.3 + 68.1i)5-s + (−100. − 81.7i)7-s + (−40.5 + 70.1i)9-s + (−345. − 598. i)11-s + 818.·13-s − 708.·15-s + (−554. − 960. i)17-s + (286. − 496. i)19-s + (184. − 1.15e3i)21-s + (−1.25e3 + 2.18e3i)23-s + (−1.53e3 − 2.65e3i)25-s − 729·27-s − 3.25e3·29-s + (−5.05e3 − 8.76e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.703 + 1.21i)5-s + (−0.776 − 0.630i)7-s + (−0.166 + 0.288i)9-s + (−0.861 − 1.49i)11-s + 1.34·13-s − 0.812·15-s + (−0.465 − 0.806i)17-s + (0.182 − 0.315i)19-s + (0.0913 − 0.570i)21-s + (−0.496 + 0.859i)23-s + (−0.490 − 0.849i)25-s − 0.192·27-s − 0.719·29-s + (−0.945 − 1.63i)31-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.580+0.814i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.580+0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.580+0.814i-0.580 + 0.814i
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), −0.580+0.814i)(2,\ 84,\ (\ :5/2),\ -0.580 + 0.814i)

Particular Values

L(3)L(3) ≈\approx 0.114845−0.222843i0.114845 - 0.222843i
L(12)L(\frac12) ≈\approx 0.114845−0.222843i0.114845 - 0.222843i
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+(100.+81.7i)T 1 + (100. + 81.7i)T
good5 1+(39.3−68.1i)T+(−1.56e3−2.70e3i)T2 1 + (39.3 - 68.1i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(345.+598.i)T+(−8.05e4+1.39e5i)T2 1 + (345. + 598. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−818.T+3.71e5T2 1 - 818.T + 3.71e5T^{2}
17 1+(554.+960.i)T+(−7.09e5+1.22e6i)T2 1 + (554. + 960. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−286.+496.i)T+(−1.23e6−2.14e6i)T2 1 + (-286. + 496. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(1.25e3−2.18e3i)T+(−3.21e6−5.57e6i)T2 1 + (1.25e3 - 2.18e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+3.25e3T+2.05e7T2 1 + 3.25e3T + 2.05e7T^{2}
31 1+(5.05e3+8.76e3i)T+(−1.43e7+2.47e7i)T2 1 + (5.05e3 + 8.76e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(2.43e3−4.21e3i)T+(−3.46e7−6.00e7i)T2 1 + (2.43e3 - 4.21e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+1.30e4T+1.15e8T2 1 + 1.30e4T + 1.15e8T^{2}
43 1+9.30e3T+1.47e8T2 1 + 9.30e3T + 1.47e8T^{2}
47 1+(6.45e3−1.11e4i)T+(−1.14e8−1.98e8i)T2 1 + (6.45e3 - 1.11e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−9.77e3−1.69e4i)T+(−2.09e8+3.62e8i)T2 1 + (-9.77e3 - 1.69e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.25e4−2.17e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.25e4 - 2.17e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.56e4+2.71e4i)T+(−4.22e8−7.31e8i)T2 1 + (-1.56e4 + 2.71e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(2.79e4+4.84e4i)T+(−6.75e8+1.16e9i)T2 1 + (2.79e4 + 4.84e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−2.05e4T+1.80e9T2 1 - 2.05e4T + 1.80e9T^{2}
73 1+(−3.38e4−5.85e4i)T+(−1.03e9+1.79e9i)T2 1 + (-3.38e4 - 5.85e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(7.03e3−1.21e4i)T+(−1.53e9−2.66e9i)T2 1 + (7.03e3 - 1.21e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+7.71e4T+3.93e9T2 1 + 7.71e4T + 3.93e9T^{2}
89 1+(160.−277.i)T+(−2.79e9−4.83e9i)T2 1 + (160. - 277. i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.12e5T+8.58e9T2 1 + 1.12e5T + 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

−13.39730578106393180371757284254, −11.25379554717136408593980789242, −10.99313555177634479167353052233, −9.720468430261817521506583853813, −8.306685128970502208157027819600, −7.16485187391012257037199814276, −5.86943257441109468115923548512, −3.75439062531395380934642240308, −3.06782495035286905844898922359, −0.097173184674663072015220257702, 1.80678864141657138763027891625, 3.76456480697270000393272163943, 5.27893519718277550736754900197, 6.80461502144275504171245742293, 8.232673749181011799991607386553, 8.884804332923566815831229099069, 10.29127135283901479309559263535, 11.93095783058076930191312645132, 12.73647252415185237504347867009, 13.17664300971680886733399317619

Graph of the ZZ-function along the critical line