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Properties

Label 2-42e2-7.2-c3-0-16
Degree 22
Conductor 17641764
Sign 0.605−0.795i0.605 - 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 − 12.1i)5-s + (2 + 3.46i)11-s + 54·13-s + (−7 − 12.1i)17-s + (−46 + 79.6i)19-s + (−76 + 131. i)23-s + (−35.5 − 61.4i)25-s + 106·29-s + (72 + 124. i)31-s + (−79 + 136. i)37-s + 390·41-s − 508·43-s + (−264 + 457. i)47-s + (303 + 524. i)53-s + 56·55-s + ⋯
L(s)  = 1  + (0.626 − 1.08i)5-s + (0.0548 + 0.0949i)11-s + 1.15·13-s + (−0.0998 − 0.172i)17-s + (−0.555 + 0.962i)19-s + (−0.689 + 1.19i)23-s + (−0.284 − 0.491i)25-s + 0.678·29-s + (0.417 + 0.722i)31-s + (−0.351 + 0.607i)37-s + 1.48·41-s − 1.80·43-s + (−0.819 + 1.41i)47-s + (0.785 + 1.36i)53-s + 0.137·55-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: 0.605−0.795i0.605 - 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 2.0756610052.075661005
L(12)L(\frac12) ≈\approx 2.0756610052.075661005
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+12.1i)T+(−62.5−108.i)T2 1 + (-7 + 12.1i)T + (-62.5 - 108. i)T^{2}
11 1+(−2−3.46i)T+(−665.5+1.15e3i)T2 1 + (-2 - 3.46i)T + (-665.5 + 1.15e3i)T^{2}
13 1−54T+2.19e3T2 1 - 54T + 2.19e3T^{2}
17 1+(7+12.1i)T+(−2.45e3+4.25e3i)T2 1 + (7 + 12.1i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(46−79.6i)T+(−3.42e3−5.94e3i)T2 1 + (46 - 79.6i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(76−131.i)T+(−6.08e3−1.05e4i)T2 1 + (76 - 131. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−106T+2.43e4T2 1 - 106T + 2.43e4T^{2}
31 1+(−72−124.i)T+(−1.48e4+2.57e4i)T2 1 + (-72 - 124. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(79−136.i)T+(−2.53e4−4.38e4i)T2 1 + (79 - 136. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−390T+6.89e4T2 1 - 390T + 6.89e4T^{2}
43 1+508T+7.95e4T2 1 + 508T + 7.95e4T^{2}
47 1+(264−457.i)T+(−5.19e4−8.99e4i)T2 1 + (264 - 457. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−303−524.i)T+(−7.44e4+1.28e5i)T2 1 + (-303 - 524. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(182+315.i)T+(−1.02e5+1.77e5i)T2 1 + (182 + 315. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(339−587.i)T+(−1.13e5−1.96e5i)T2 1 + (339 - 587. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(422+730.i)T+(−1.50e5+2.60e5i)T2 1 + (422 + 730. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−8T+3.57e5T2 1 - 8T + 3.57e5T^{2}
73 1+(−211−365.i)T+(−1.94e5+3.36e5i)T2 1 + (-211 - 365. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(192−332.i)T+(−2.46e5−4.26e5i)T2 1 + (192 - 332. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−548T+5.71e5T2 1 - 548T + 5.71e5T^{2}
89 1+(−597+1.03e3i)T+(−3.52e5−6.10e5i)T2 1 + (-597 + 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+1.50e3T+9.12e5T2 1 + 1.50e3T + 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

−9.000180299755715829088376851785, −8.390050338117211424068514403469, −7.66176092919759085609032099807, −6.39334461444201393752614273754, −5.89016383269228181608208425200, −4.98732680325032748249532463194, −4.17001431250774838681421756755, −3.14596983837836529175548715100, −1.71262069853415783459814326454, −1.14173216445116802085574670016, 0.44946874433453663507770217237, 1.91467595197289096487872979518, 2.71661798743668879415446923587, 3.69206658365482886912808801825, 4.66046360095585425227876365261, 5.85178418090096910570265142813, 6.46381411334078012928968298372, 6.95532128416389802725733227029, 8.187958782572275638078661158058, 8.716513940347302091798096182462

Graph of the ZZ-function along the critical line