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Properties

Label 2-6300-21.17-c1-0-17
Degree 22
Conductor 63006300
Sign 0.999−0.0243i0.999 - 0.0243i
Analytic cond. 50.305750.3057
Root an. cond. 7.092657.09265
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  + (−1.29 − 2.30i)7-s + (−1.33 − 0.768i)11-s − 0.219i·13-s + (0.713 − 1.23i)17-s + (−3.80 + 2.19i)19-s + (−0.673 + 0.389i)23-s + 6.82i·29-s + (5.32 + 3.07i)31-s + (2.99 + 5.19i)37-s − 8.90·41-s − 1.91·43-s + (5.60 + 9.70i)47-s + (−3.62 + 5.98i)49-s + (4.18 + 2.41i)53-s + (0.336 − 0.582i)59-s + ⋯
L(s)  = 1  + (−0.490 − 0.871i)7-s + (−0.401 − 0.231i)11-s − 0.0608i·13-s + (0.172 − 0.299i)17-s + (−0.872 + 0.503i)19-s + (−0.140 + 0.0811i)23-s + 1.26i·29-s + (0.956 + 0.552i)31-s + (0.492 + 0.853i)37-s − 1.39·41-s − 0.292·43-s + (0.817 + 1.41i)47-s + (−0.518 + 0.855i)49-s + (0.574 + 0.331i)53-s + (0.0437 − 0.0758i)59-s + ⋯

Functional equation

Λ(s)=(6300s/2 ΓC(s) L(s)=((0.999−0.0243i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0243i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6300s/2 ΓC(s+1/2) L(s)=((0.999−0.0243i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0243i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63006300    =    22⋅32⋅52⋅72^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7
Sign: 0.999−0.0243i0.999 - 0.0243i
Analytic conductor: 50.305750.3057
Root analytic conductor: 7.092657.09265
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6300(4301,⋅)\chi_{6300} (4301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6300, ( :1/2), 0.999−0.0243i)(2,\ 6300,\ (\ :1/2),\ 0.999 - 0.0243i)

Particular Values

L(1)L(1) ≈\approx 1.4792749961.479274996
L(12)L(\frac12) ≈\approx 1.4792749961.479274996
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)
bad2 1 1
3 1 1
5 1 1
7 1+(1.29+2.30i)T 1 + (1.29 + 2.30i)T
good11 1+(1.33+0.768i)T+(5.5+9.52i)T2 1 + (1.33 + 0.768i)T + (5.5 + 9.52i)T^{2}
13 1+0.219iT−13T2 1 + 0.219iT - 13T^{2}
17 1+(−0.713+1.23i)T+(−8.5−14.7i)T2 1 + (-0.713 + 1.23i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.80−2.19i)T+(9.5−16.4i)T2 1 + (3.80 - 2.19i)T + (9.5 - 16.4i)T^{2}
23 1+(0.673−0.389i)T+(11.5−19.9i)T2 1 + (0.673 - 0.389i)T + (11.5 - 19.9i)T^{2}
29 1−6.82iT−29T2 1 - 6.82iT - 29T^{2}
31 1+(−5.32−3.07i)T+(15.5+26.8i)T2 1 + (-5.32 - 3.07i)T + (15.5 + 26.8i)T^{2}
37 1+(−2.99−5.19i)T+(−18.5+32.0i)T2 1 + (-2.99 - 5.19i)T + (-18.5 + 32.0i)T^{2}
41 1+8.90T+41T2 1 + 8.90T + 41T^{2}
43 1+1.91T+43T2 1 + 1.91T + 43T^{2}
47 1+(−5.60−9.70i)T+(−23.5+40.7i)T2 1 + (-5.60 - 9.70i)T + (-23.5 + 40.7i)T^{2}
53 1+(−4.18−2.41i)T+(26.5+45.8i)T2 1 + (-4.18 - 2.41i)T + (26.5 + 45.8i)T^{2}
59 1+(−0.336+0.582i)T+(−29.5−51.0i)T2 1 + (-0.336 + 0.582i)T + (-29.5 - 51.0i)T^{2}
61 1+(−9.64+5.56i)T+(30.5−52.8i)T2 1 + (-9.64 + 5.56i)T + (30.5 - 52.8i)T^{2}
67 1+(−5.65+9.78i)T+(−33.5−58.0i)T2 1 + (-5.65 + 9.78i)T + (-33.5 - 58.0i)T^{2}
71 1+8.48iT−71T2 1 + 8.48iT - 71T^{2}
73 1+(1.88+1.09i)T+(36.5+63.2i)T2 1 + (1.88 + 1.09i)T + (36.5 + 63.2i)T^{2}
79 1+(3.19+5.53i)T+(−39.5+68.4i)T2 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2}
83 1−5.09T+83T2 1 - 5.09T + 83T^{2}
89 1+(−2.97−5.15i)T+(−44.5+77.0i)T2 1 + (-2.97 - 5.15i)T + (-44.5 + 77.0i)T^{2}
97 1+12.7iT−97T2 1 + 12.7iT - 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

−8.045316471348587623235933313779, −7.31235199708929248191474972765, −6.62290696892860848501952110115, −6.07419137165348865738741243130, −5.07331739909457531871898347982, −4.46464376363594717681463285337, −3.51350700969224775283842567561, −2.96392243907489154058744596315, −1.76576306452209620335062193956, −0.68193502591586643432370889422, 0.56195921557517308292363659846, 2.15212639862435527424749403059, 2.52574583326005004876145772741, 3.65737782756810352240011004840, 4.37121246643170749038605223624, 5.31236540424683469184128419614, 5.87833676679504510389507180547, 6.61364659594462110402256866891, 7.23515154453276075437223502110, 8.317087028104678643426840207016

Graph of the ZZ-function along the critical line