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Properties

Label 2-21e2-7.4-c3-0-43
Degree 22
Conductor 441441
Sign −0.991+0.126i-0.991 + 0.126i
Analytic cond. 26.019826.0198
Root an. cond. 5.100965.10096
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  + (4 − 6.92i)4-s − 70·13-s + (−31.9 − 55.4i)16-s + (−28 − 48.4i)19-s + (62.5 − 108. i)25-s + (−154 + 266. i)31-s + (−55 − 95.2i)37-s − 520·43-s + (−280 + 484. i)52-s + (−91 − 157. i)61-s − 511.·64-s + (440 − 762. i)67-s + (−595 + 1.03e3i)73-s − 448·76-s + (−442 − 765. i)79-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s − 1.49·13-s + (−0.499 − 0.866i)16-s + (−0.338 − 0.585i)19-s + (0.5 − 0.866i)25-s + (−0.892 + 1.54i)31-s + (−0.244 − 0.423i)37-s − 1.84·43-s + (−0.746 + 1.29i)52-s + (−0.191 − 0.330i)61-s − 0.999·64-s + (0.802 − 1.38i)67-s + (−0.953 + 1.65i)73-s − 0.676·76-s + (−0.629 − 1.09i)79-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+3/2) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.991+0.126i-0.991 + 0.126i
Analytic conductor: 26.019826.0198
Root analytic conductor: 5.100965.10096
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ441(361,⋅)\chi_{441} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :3/2), −0.991+0.126i)(2,\ 441,\ (\ :3/2),\ -0.991 + 0.126i)

Particular Values

L(2)L(2) ≈\approx 0.73861728220.7386172822
L(12)L(\frac12) ≈\approx 0.73861728220.7386172822
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)
bad3 1 1
7 1 1
good2 1+(−4+6.92i)T2 1 + (-4 + 6.92i)T^{2}
5 1+(−62.5+108.i)T2 1 + (-62.5 + 108. i)T^{2}
11 1+(−665.5−1.15e3i)T2 1 + (-665.5 - 1.15e3i)T^{2}
13 1+70T+2.19e3T2 1 + 70T + 2.19e3T^{2}
17 1+(−2.45e3−4.25e3i)T2 1 + (-2.45e3 - 4.25e3i)T^{2}
19 1+(28+48.4i)T+(−3.42e3+5.94e3i)T2 1 + (28 + 48.4i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−6.08e3+1.05e4i)T2 1 + (-6.08e3 + 1.05e4i)T^{2}
29 1+2.43e4T2 1 + 2.43e4T^{2}
31 1+(154−266.i)T+(−1.48e4−2.57e4i)T2 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(55+95.2i)T+(−2.53e4+4.38e4i)T2 1 + (55 + 95.2i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+6.89e4T2 1 + 6.89e4T^{2}
43 1+520T+7.95e4T2 1 + 520T + 7.95e4T^{2}
47 1+(−5.19e4+8.99e4i)T2 1 + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−7.44e4−1.28e5i)T2 1 + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−1.02e5−1.77e5i)T2 1 + (-1.02e5 - 1.77e5i)T^{2}
61 1+(91+157.i)T+(−1.13e5+1.96e5i)T2 1 + (91 + 157. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−440+762.i)T+(−1.50e5−2.60e5i)T2 1 + (-440 + 762. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1+(595−1.03e3i)T+(−1.94e5−3.36e5i)T2 1 + (595 - 1.03e3i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(442+765.i)T+(−2.46e5+4.26e5i)T2 1 + (442 + 765. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+5.71e5T2 1 + 5.71e5T^{2}
89 1+(−3.52e5+6.10e5i)T2 1 + (-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

−10.27956083209803175970728823408, −9.567272421880884948901266038398, −8.529049672316125847966618328334, −7.22313989641582163104713892595, −6.64668630759271982649604512118, −5.39489699364415509355371136752, −4.65829605843546070449210196188, −2.92776921553246232766972661783, −1.80493315745799743006436770142, −0.21407224702063321874449893990, 1.94842047405757395210193268107, 3.05964471367929791532014632990, 4.20949255296742531605535240181, 5.41828436421701897590208947581, 6.70967973899054809091711450815, 7.45779498421284055779773259075, 8.251388036442077242208349159031, 9.332757193244271912365143230436, 10.25081169568286480137101051612, 11.30453194257185347936891286881

Graph of the ZZ-function along the critical line