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Properties

Label 2-42e2-7.2-c3-0-1
Degree 22
Conductor 17641764
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 104.079104.079
Root an. cond. 10.201910.2019
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.93 + 13.7i)5-s + (−7.93 − 13.7i)11-s + 26·13-s + (39.6 + 68.7i)17-s + (−34 + 58.8i)19-s + (−23.8 + 41.2i)23-s + (−63.5 − 109. i)25-s − 253.·29-s + (−106 − 183. i)31-s + (−109 + 188. i)37-s + 396.·41-s + 260·43-s + (−206. + 357. i)47-s + (238. + 412. i)53-s + 252·55-s + ⋯
L(s)  = 1  + (−0.709 + 1.22i)5-s + (−0.217 − 0.376i)11-s + 0.554·13-s + (0.566 + 0.980i)17-s + (−0.410 + 0.711i)19-s + (−0.215 + 0.373i)23-s + (−0.508 − 0.879i)25-s − 1.62·29-s + (−0.614 − 1.06i)31-s + (−0.484 + 0.838i)37-s + 1.51·41-s + 0.922·43-s + (−0.640 + 1.10i)47-s + (0.617 + 1.06i)53-s + 0.617·55-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+3/2) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 104.079104.079
Root analytic conductor: 10.201910.2019
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1764(1549,⋅)\chi_{1764} (1549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :3/2), −0.605+0.795i)(2,\ 1764,\ (\ :3/2),\ -0.605 + 0.795i)

Particular Values

L(2)L(2) ≈\approx 0.21874772760.2187477276
L(12)L(\frac12) ≈\approx 0.21874772760.2187477276
L(52)L(\frac{5}{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 1
7 1 1
good5 1+(7.93−13.7i)T+(−62.5−108.i)T2 1 + (7.93 - 13.7i)T + (-62.5 - 108. i)T^{2}
11 1+(7.93+13.7i)T+(−665.5+1.15e3i)T2 1 + (7.93 + 13.7i)T + (-665.5 + 1.15e3i)T^{2}
13 1−26T+2.19e3T2 1 - 26T + 2.19e3T^{2}
17 1+(−39.6−68.7i)T+(−2.45e3+4.25e3i)T2 1 + (-39.6 - 68.7i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(34−58.8i)T+(−3.42e3−5.94e3i)T2 1 + (34 - 58.8i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(23.8−41.2i)T+(−6.08e3−1.05e4i)T2 1 + (23.8 - 41.2i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+253.T+2.43e4T2 1 + 253.T + 2.43e4T^{2}
31 1+(106+183.i)T+(−1.48e4+2.57e4i)T2 1 + (106 + 183. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(109−188.i)T+(−2.53e4−4.38e4i)T2 1 + (109 - 188. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−396.T+6.89e4T2 1 - 396.T + 6.89e4T^{2}
43 1−260T+7.95e4T2 1 - 260T + 7.95e4T^{2}
47 1+(206.−357.i)T+(−5.19e4−8.99e4i)T2 1 + (206. - 357. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−238.−412.i)T+(−7.44e4+1.28e5i)T2 1 + (-238. - 412. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−142.−247.i)T+(−1.02e5+1.77e5i)T2 1 + (-142. - 247. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−161+278.i)T+(−1.13e5−1.96e5i)T2 1 + (-161 + 278. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(178+308.i)T+(−1.50e5+2.60e5i)T2 1 + (178 + 308. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+1.12e3T+3.57e5T2 1 + 1.12e3T + 3.57e5T^{2}
73 1+(−113−195.i)T+(−1.94e5+3.36e5i)T2 1 + (-113 - 195. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(220−381.i)T+(−2.46e5−4.26e5i)T2 1 + (220 - 381. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+253.T+5.71e5T2 1 + 253.T + 5.71e5T^{2}
89 1+(103.−178.i)T+(−3.52e5−6.10e5i)T2 1 + (103. - 178. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+1.33e3T+9.12e5T2 1 + 1.33e3T + 9.12e5T^{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

−9.510435576614936291782344521212, −8.501562217958942000813103749307, −7.71730073675868204537974338563, −7.28807492185824103439571408293, −6.07127805331913061246402506489, −5.78833047167547811424385117159, −4.15514592501247466465426756013, −3.64785516421569821366821487357, −2.74358864414720114985497165604, −1.52030078597777683165342806973, 0.05471439942842974204613360136, 0.945436152918830334699302371880, 2.16536164109119294067795391576, 3.49146064558070301311141666565, 4.30900669077086508541689099467, 5.09382990306391556786280602826, 5.78641377561631990616022013055, 7.10830177178368523224722329755, 7.58768074611974818855962747304, 8.630243708809585956291887983248

Graph of the ZZ-function along the critical line