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Properties

Label 2-21e2-7.4-c1-0-7
Degree 22
Conductor 441441
Sign −0.991+0.126i-0.991 + 0.126i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 2.29i)2-s + (−2.5 + 4.33i)4-s + 7.93·8-s + (2.64 − 4.58i)11-s + (−5.49 − 9.52i)16-s − 14·22-s + (−2.64 − 4.58i)23-s + (2.5 − 4.33i)25-s − 10.5·29-s + (−6.61 + 11.4i)32-s + (−3 − 5.19i)37-s + 12·43-s + (13.2 + 22.9i)44-s + (−7 + 12.1i)46-s − 13.2·50-s + ⋯
L(s)  = 1  + (−0.935 − 1.62i)2-s + (−1.25 + 2.16i)4-s + 2.80·8-s + (0.797 − 1.38i)11-s + (−1.37 − 2.38i)16-s − 2.98·22-s + (−0.551 − 0.955i)23-s + (0.5 − 0.866i)25-s − 1.96·29-s + (−1.16 + 2.02i)32-s + (−0.493 − 0.854i)37-s + 1.82·43-s + (1.99 + 3.45i)44-s + (−1.03 + 1.78i)46-s − 1.87·50-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(361,⋅)\chi_{441} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), −0.991+0.126i)(2,\ 441,\ (\ :1/2),\ -0.991 + 0.126i)

Particular Values

L(1)L(1) ≈\approx 0.0431640−0.680175i0.0431640 - 0.680175i
L(12)L(\frac12) ≈\approx 0.0431640−0.680175i0.0431640 - 0.680175i
L(32)L(\frac{3}{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+(1.32+2.29i)T+(−1+1.73i)T2 1 + (1.32 + 2.29i)T + (-1 + 1.73i)T^{2}
5 1+(−2.5+4.33i)T2 1 + (-2.5 + 4.33i)T^{2}
11 1+(−2.64+4.58i)T+(−5.5−9.52i)T2 1 + (-2.64 + 4.58i)T + (-5.5 - 9.52i)T^{2}
13 1+13T2 1 + 13T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(2.64+4.58i)T+(−11.5+19.9i)T2 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2}
29 1+10.5T+29T2 1 + 10.5T + 29T^{2}
31 1+(−15.5−26.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1+(3+5.19i)T+(−18.5+32.0i)T2 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2}
41 1+41T2 1 + 41T^{2}
43 1−12T+43T2 1 - 12T + 43T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1+(−5.29+9.16i)T+(−26.5−45.8i)T2 1 + (-5.29 + 9.16i)T + (-26.5 - 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(2−3.46i)T+(−33.5−58.0i)T2 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1+5.29T+71T2 1 + 5.29T + 71T^{2}
73 1+(−36.5−63.2i)T2 1 + (-36.5 - 63.2i)T^{2}
79 1+(4+6.92i)T+(−39.5+68.4i)T2 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−44.5+77.0i)T2 1 + (-44.5 + 77.0i)T^{2}
97 1+97T2 1 + 97T^{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

−10.82609173296786453330035239923, −9.911812943306868928725525944038, −8.943031227210958481236175773362, −8.516195764939302457246275916807, −7.37985377558115264145451324182, −5.92317704149626451149910777062, −4.23004013721510196810552384000, −3.38024253899097073966283710723, −2.14009531752356935003789463378, −0.62738357114098857897071554810, 1.57524628942162217482476555325, 4.10954644861899663574982754642, 5.25115218957661253206508273131, 6.15070334950467214322174731646, 7.26599583516057262854233745818, 7.53675227555732028248516407599, 8.892251051294420503655452830306, 9.412375924765196334903300457434, 10.16521409613907406703997374738, 11.30181091589250696727956659177

Graph of the ZZ-function along the critical line