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Properties

Label 2-224-7.4-c1-0-3
Degree 22
Conductor 224224
Sign 0.827−0.561i0.827 - 0.561i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
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  + (−0.207 + 0.358i)3-s + (0.914 + 1.58i)5-s + (1 − 2.44i)7-s + (1.41 + 2.44i)9-s + (−1.20 + 2.09i)11-s + 2.82·13-s − 0.757·15-s + (0.0857 − 0.148i)17-s + (3.20 + 5.55i)19-s + (0.671 + 0.866i)21-s + (−2.62 − 4.54i)23-s + (0.828 − 1.43i)25-s − 2.41·27-s − 2.82·29-s + (2.79 − 4.83i)31-s + ⋯
L(s)  = 1  + (−0.119 + 0.207i)3-s + (0.408 + 0.708i)5-s + (0.377 − 0.925i)7-s + (0.471 + 0.816i)9-s + (−0.363 + 0.630i)11-s + 0.784·13-s − 0.195·15-s + (0.0208 − 0.0360i)17-s + (0.735 + 1.27i)19-s + (0.146 + 0.188i)21-s + (−0.546 − 0.946i)23-s + (0.165 − 0.286i)25-s − 0.464·27-s − 0.525·29-s + (0.501 − 0.868i)31-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.827−0.561i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.827−0.561i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.827−0.561i0.827 - 0.561i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(193,⋅)\chi_{224} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.827−0.561i)(2,\ 224,\ (\ :1/2),\ 0.827 - 0.561i)

Particular Values

L(1)L(1) ≈\approx 1.26118+0.387524i1.26118 + 0.387524i
L(12)L(\frac12) ≈\approx 1.26118+0.387524i1.26118 + 0.387524i
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
7 1+(−1+2.44i)T 1 + (-1 + 2.44i)T
good3 1+(0.207−0.358i)T+(−1.5−2.59i)T2 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2}
5 1+(−0.914−1.58i)T+(−2.5+4.33i)T2 1 + (-0.914 - 1.58i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.20−2.09i)T+(−5.5−9.52i)T2 1 + (1.20 - 2.09i)T + (-5.5 - 9.52i)T^{2}
13 1−2.82T+13T2 1 - 2.82T + 13T^{2}
17 1+(−0.0857+0.148i)T+(−8.5−14.7i)T2 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.20−5.55i)T+(−9.5+16.4i)T2 1 + (-3.20 - 5.55i)T + (-9.5 + 16.4i)T^{2}
23 1+(2.62+4.54i)T+(−11.5+19.9i)T2 1 + (2.62 + 4.54i)T + (-11.5 + 19.9i)T^{2}
29 1+2.82T+29T2 1 + 2.82T + 29T^{2}
31 1+(−2.79+4.83i)T+(−15.5−26.8i)T2 1 + (-2.79 + 4.83i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.32+7.49i)T+(−18.5+32.0i)T2 1 + (4.32 + 7.49i)T + (-18.5 + 32.0i)T^{2}
41 1+6.82T+41T2 1 + 6.82T + 41T^{2}
43 1+9.65T+43T2 1 + 9.65T + 43T^{2}
47 1+(−5.20−9.01i)T+(−23.5+40.7i)T2 1 + (-5.20 - 9.01i)T + (-23.5 + 40.7i)T^{2}
53 1+(−0.5+0.866i)T+(−26.5−45.8i)T2 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2}
59 1+(−5.44+9.43i)T+(−29.5−51.0i)T2 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.32+7.49i)T+(−30.5+52.8i)T2 1 + (4.32 + 7.49i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.37−2.38i)T+(−33.5−58.0i)T2 1 + (1.37 - 2.38i)T + (-33.5 - 58.0i)T^{2}
71 1+13.6T+71T2 1 + 13.6T + 71T^{2}
73 1+(−7.32+12.6i)T+(−36.5−63.2i)T2 1 + (-7.32 + 12.6i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.03−5.25i)T+(−39.5+68.4i)T2 1 + (-3.03 - 5.25i)T + (-39.5 + 68.4i)T^{2}
83 1−7.31T+83T2 1 - 7.31T + 83T^{2}
89 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+1.17T+97T2 1 + 1.17T + 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

−12.35806154393588640752178806093, −11.08176790504918435962069633158, −10.40073985683443265315842389783, −9.843491944862121242722525960421, −8.163281339936845850747127716014, −7.38496135257613797120916587932, −6.25108979036604278242190490554, −4.92117242975244540115233052290, −3.73839530720966348506545883642, −1.93320062019993050476359583201, 1.42020400044126229265447745761, 3.29620020794666943077395586527, 5.01268817516358308506779998655, 5.82999175809722255806958740562, 7.03195584955686508293638979219, 8.501176235773568457284546710126, 9.029987772690501658736546380094, 10.12723512570474288309290752171, 11.51839685059725683568442794199, 12.02151322567268289742403550713

Graph of the ZZ-function along the critical line