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Properties

Label 2-2061-229.177-c0-0-0
Degree 22
Conductor 20612061
Sign 0.705+0.708i0.705 + 0.708i
Analytic cond. 1.028571.02857
Root an. cond. 1.014181.01418
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.614 − 0.789i)4-s + (0.0769 + 0.0300i)7-s + (0.981 + 0.292i)13-s + (−0.245 + 0.969i)16-s + (0.706 + 0.382i)19-s + (0.546 − 0.837i)25-s + (−0.0235 − 0.0792i)28-s + (0.986 − 1.16i)31-s + (−0.484 − 1.91i)37-s + (0.159 + 0.629i)43-s + (−0.730 − 0.672i)49-s + (−0.372 − 0.953i)52-s + (−1.39 − 0.477i)61-s + (0.915 − 0.401i)64-s + (0.512 − 0.250i)67-s + ⋯
L(s)  = 1  + (−0.614 − 0.789i)4-s + (0.0769 + 0.0300i)7-s + (0.981 + 0.292i)13-s + (−0.245 + 0.969i)16-s + (0.706 + 0.382i)19-s + (0.546 − 0.837i)25-s + (−0.0235 − 0.0792i)28-s + (0.986 − 1.16i)31-s + (−0.484 − 1.91i)37-s + (0.159 + 0.629i)43-s + (−0.730 − 0.672i)49-s + (−0.372 − 0.953i)52-s + (−1.39 − 0.477i)61-s + (0.915 − 0.401i)64-s + (0.512 − 0.250i)67-s + ⋯

Functional equation

Λ(s)=(2061s/2 ΓC(s) L(s)=((0.705+0.708i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2061s/2 ΓC(s) L(s)=((0.705+0.708i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20612061    =    32⋅2293^{2} \cdot 229
Sign: 0.705+0.708i0.705 + 0.708i
Analytic conductor: 1.028571.02857
Root analytic conductor: 1.014181.01418
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2061(406,⋅)\chi_{2061} (406, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2061, ( :0), 0.705+0.708i)(2,\ 2061,\ (\ :0),\ 0.705 + 0.708i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0604525031.060452503
L(12)L(\frac12) ≈\approx 1.0604525031.060452503
L(1)L(1) 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
229 1+(0.735+0.677i)T 1 + (0.735 + 0.677i)T
good2 1+(0.614+0.789i)T2 1 + (0.614 + 0.789i)T^{2}
5 1+(−0.546+0.837i)T2 1 + (-0.546 + 0.837i)T^{2}
7 1+(−0.0769−0.0300i)T+(0.735+0.677i)T2 1 + (-0.0769 - 0.0300i)T + (0.735 + 0.677i)T^{2}
11 1+(0.986−0.164i)T2 1 + (0.986 - 0.164i)T^{2}
13 1+(−0.981−0.292i)T+(0.837+0.546i)T2 1 + (-0.981 - 0.292i)T + (0.837 + 0.546i)T^{2}
17 1+(0.546−0.837i)T2 1 + (0.546 - 0.837i)T^{2}
19 1+(−0.706−0.382i)T+(0.546+0.837i)T2 1 + (-0.706 - 0.382i)T + (0.546 + 0.837i)T^{2}
23 1+(−0.996−0.0825i)T2 1 + (-0.996 - 0.0825i)T^{2}
29 1+(−0.735−0.677i)T2 1 + (-0.735 - 0.677i)T^{2}
31 1+(−0.986+1.16i)T+(−0.164−0.986i)T2 1 + (-0.986 + 1.16i)T + (-0.164 - 0.986i)T^{2}
37 1+(0.484+1.91i)T+(−0.879+0.475i)T2 1 + (0.484 + 1.91i)T + (-0.879 + 0.475i)T^{2}
41 1+(−0.614−0.789i)T2 1 + (-0.614 - 0.789i)T^{2}
43 1+(−0.159−0.629i)T+(−0.879+0.475i)T2 1 + (-0.159 - 0.629i)T + (-0.879 + 0.475i)T^{2}
47 1+(0.614−0.789i)T2 1 + (0.614 - 0.789i)T^{2}
53 1+(0.945+0.324i)T2 1 + (0.945 + 0.324i)T^{2}
59 1+(0.475−0.879i)T2 1 + (0.475 - 0.879i)T^{2}
61 1+(1.39+0.477i)T+(0.789+0.614i)T2 1 + (1.39 + 0.477i)T + (0.789 + 0.614i)T^{2}
67 1+(−0.512+0.250i)T+(0.614−0.789i)T2 1 + (-0.512 + 0.250i)T + (0.614 - 0.789i)T^{2}
71 1+(0.986+0.164i)T2 1 + (0.986 + 0.164i)T^{2}
73 1+(−1.59+0.334i)T+(0.915−0.401i)T2 1 + (-1.59 + 0.334i)T + (0.915 - 0.401i)T^{2}
79 1+(−0.711−1.82i)T+(−0.735+0.677i)T2 1 + (-0.711 - 1.82i)T + (-0.735 + 0.677i)T^{2}
83 1+(−0.879−0.475i)T2 1 + (-0.879 - 0.475i)T^{2}
89 1+iT2 1 + iT^{2}
97 1+(1.62−1.06i)T+(0.401−0.915i)T2 1 + (1.62 - 1.06i)T + (0.401 - 0.915i)T^{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

−9.324677516593207663788380522171, −8.495176827509102513198757763459, −7.84476346303341139680765399295, −6.65627952986479550338022312281, −6.03149126804027703935120545044, −5.24724995762836450013459628976, −4.36828486555592519040403113333, −3.56930294240008492587227167776, −2.16470389066727843652982616073, −0.955127292869200122622876721431, 1.25942301374903281258607465362, 2.95884584301470603633934984340, 3.47884118252817192165470821090, 4.60367509535479472943173193013, 5.21951415061908316790043944694, 6.35439954548719790968838416283, 7.16141447257839856059612313622, 8.002934709869528584476099360380, 8.605870827615751940638000106344, 9.232159599135928982118932738209

Graph of the ZZ-function along the critical line