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Properties

Label 2-84-7.2-c5-0-6
Degree 22
Conductor 8484
Sign −0.349+0.937i-0.349 + 0.937i
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 + (46.4 − 80.3i)5-s + (−118. − 52.8i)7-s + (−40.5 + 70.1i)9-s + (−70.3 − 121. i)11-s − 1.11e3·13-s + 835.·15-s + (−27.4 − 47.5i)17-s + (855. − 1.48e3i)19-s + (−120. − 1.16e3i)21-s + (1.64e3 − 2.84e3i)23-s + (−2.74e3 − 4.75e3i)25-s − 729·27-s − 3.79e3·29-s + (2.42e3 + 4.19e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.830 − 1.43i)5-s + (−0.913 − 0.407i)7-s + (−0.166 + 0.288i)9-s + (−0.175 − 0.303i)11-s − 1.82·13-s + 0.958·15-s + (−0.0230 − 0.0398i)17-s + (0.543 − 0.942i)19-s + (−0.0596 − 0.574i)21-s + (0.647 − 1.12i)23-s + (−0.878 − 1.52i)25-s − 0.192·27-s − 0.837·29-s + (0.452 + 0.784i)31-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.349+0.937i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.349+0.937i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.349+0.937i-0.349 + 0.937i
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), −0.349+0.937i)(2,\ 84,\ (\ :5/2),\ -0.349 + 0.937i)

Particular Values

L(3)L(3) ≈\approx 0.785765−1.13138i0.785765 - 1.13138i
L(12)L(\frac12) ≈\approx 0.785765−1.13138i0.785765 - 1.13138i
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+(118.+52.8i)T 1 + (118. + 52.8i)T
good5 1+(−46.4+80.3i)T+(−1.56e3−2.70e3i)T2 1 + (-46.4 + 80.3i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(70.3+121.i)T+(−8.05e4+1.39e5i)T2 1 + (70.3 + 121. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+1.11e3T+3.71e5T2 1 + 1.11e3T + 3.71e5T^{2}
17 1+(27.4+47.5i)T+(−7.09e5+1.22e6i)T2 1 + (27.4 + 47.5i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−855.+1.48e3i)T+(−1.23e6−2.14e6i)T2 1 + (-855. + 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−1.64e3+2.84e3i)T+(−3.21e6−5.57e6i)T2 1 + (-1.64e3 + 2.84e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+3.79e3T+2.05e7T2 1 + 3.79e3T + 2.05e7T^{2}
31 1+(−2.42e3−4.19e3i)T+(−1.43e7+2.47e7i)T2 1 + (-2.42e3 - 4.19e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−5.68e3+9.84e3i)T+(−3.46e7−6.00e7i)T2 1 + (-5.68e3 + 9.84e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+1.03e4T+1.15e8T2 1 + 1.03e4T + 1.15e8T^{2}
43 1−7.13e3T+1.47e8T2 1 - 7.13e3T + 1.47e8T^{2}
47 1+(−8.20e3+1.42e4i)T+(−1.14e8−1.98e8i)T2 1 + (-8.20e3 + 1.42e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.04e4−1.81e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.04e4 - 1.81e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.81e4−3.13e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.81e4 - 3.13e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(2.47e3−4.28e3i)T+(−4.22e8−7.31e8i)T2 1 + (2.47e3 - 4.28e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−1.14e4−1.98e4i)T+(−6.75e8+1.16e9i)T2 1 + (-1.14e4 - 1.98e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+2.63e4T+1.80e9T2 1 + 2.63e4T + 1.80e9T^{2}
73 1+(2.76e4+4.79e4i)T+(−1.03e9+1.79e9i)T2 1 + (2.76e4 + 4.79e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−2.49e4+4.32e4i)T+(−1.53e9−2.66e9i)T2 1 + (-2.49e4 + 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−4.48e4T+3.93e9T2 1 - 4.48e4T + 3.93e9T^{2}
89 1+(6.39e4−1.10e5i)T+(−2.79e9−4.83e9i)T2 1 + (6.39e4 - 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+6.56e4T+8.58e9T2 1 + 6.56e4T + 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

−13.02741430687984908008012757150, −12.19068378761526755133923235874, −10.41363733177450411376342760542, −9.504488548281649637739702314764, −8.842014451151671326518149548054, −7.19406879819791740366265629200, −5.49814810114195602304991809387, −4.51771336869341740947453197125, −2.60358281801876229093346292117, −0.51937325190237593135441804386, 2.20108013975625415349399278912, 3.16434361400646930423604786594, 5.60434048375190981678367947790, 6.75979001434597726818984506862, 7.59296479946980574305069520869, 9.610602569151661042152575981513, 9.971235118039912962144125961376, 11.54273966427230089146447707697, 12.68579488606645178240679712813, 13.67441992725665850545295592024

Graph of the ZZ-function along the critical line