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Properties

Label 2-550-55.52-c1-0-6
Degree 22
Conductor 550550
Sign 0.568+0.822i0.568 + 0.822i
Analytic cond. 4.391774.39177
Root an. cond. 2.095652.09565
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  + (−0.156 + 0.987i)2-s + (−2.86 − 1.45i)3-s + (−0.951 − 0.309i)4-s + (1.88 − 2.59i)6-s + (0.692 + 1.35i)7-s + (0.453 − 0.891i)8-s + (4.29 + 5.91i)9-s + (−2.70 + 1.91i)11-s + (2.27 + 2.27i)12-s + (−1.04 − 0.165i)13-s + (−1.44 + 0.471i)14-s + (0.809 + 0.587i)16-s + (2.19 − 0.347i)17-s + (−6.51 + 3.31i)18-s + (−1.91 − 5.89i)19-s + ⋯
L(s)  = 1  + (−0.110 + 0.698i)2-s + (−1.65 − 0.841i)3-s + (−0.475 − 0.154i)4-s + (0.770 − 1.06i)6-s + (0.261 + 0.513i)7-s + (0.160 − 0.315i)8-s + (1.43 + 1.97i)9-s + (−0.815 + 0.578i)11-s + (0.655 + 0.655i)12-s + (−0.289 − 0.0459i)13-s + (−0.387 + 0.125i)14-s + (0.202 + 0.146i)16-s + (0.532 − 0.0843i)17-s + (−1.53 + 0.782i)18-s + (−0.439 − 1.35i)19-s + ⋯

Functional equation

Λ(s)=(550s/2 ΓC(s) L(s)=((0.568+0.822i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(550s/2 ΓC(s+1/2) L(s)=((0.568+0.822i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 550550    =    2⋅52⋅112 \cdot 5^{2} \cdot 11
Sign: 0.568+0.822i0.568 + 0.822i
Analytic conductor: 4.391774.39177
Root analytic conductor: 2.095652.09565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ550(107,⋅)\chi_{550} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 550, ( :1/2), 0.568+0.822i)(2,\ 550,\ (\ :1/2),\ 0.568 + 0.822i)

Particular Values

L(1)L(1) ≈\approx 0.472159−0.247770i0.472159 - 0.247770i
L(12)L(\frac12) ≈\approx 0.472159−0.247770i0.472159 - 0.247770i
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+(0.156−0.987i)T 1 + (0.156 - 0.987i)T
5 1 1
11 1+(2.70−1.91i)T 1 + (2.70 - 1.91i)T
good3 1+(2.86+1.45i)T+(1.76+2.42i)T2 1 + (2.86 + 1.45i)T + (1.76 + 2.42i)T^{2}
7 1+(−0.692−1.35i)T+(−4.11+5.66i)T2 1 + (-0.692 - 1.35i)T + (-4.11 + 5.66i)T^{2}
13 1+(1.04+0.165i)T+(12.3+4.01i)T2 1 + (1.04 + 0.165i)T + (12.3 + 4.01i)T^{2}
17 1+(−2.19+0.347i)T+(16.1−5.25i)T2 1 + (-2.19 + 0.347i)T + (16.1 - 5.25i)T^{2}
19 1+(1.91+5.89i)T+(−15.3+11.1i)T2 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2}
23 1+(−3.05+3.05i)T−23iT2 1 + (-3.05 + 3.05i)T - 23iT^{2}
29 1+(−1.39+4.30i)T+(−23.4−17.0i)T2 1 + (-1.39 + 4.30i)T + (-23.4 - 17.0i)T^{2}
31 1+(−2.32+1.69i)T+(9.57−29.4i)T2 1 + (-2.32 + 1.69i)T + (9.57 - 29.4i)T^{2}
37 1+(1.24−0.631i)T+(21.7−29.9i)T2 1 + (1.24 - 0.631i)T + (21.7 - 29.9i)T^{2}
41 1+(−2.38+0.774i)T+(33.1−24.0i)T2 1 + (-2.38 + 0.774i)T + (33.1 - 24.0i)T^{2}
43 1+(6.40+6.40i)T+43iT2 1 + (6.40 + 6.40i)T + 43iT^{2}
47 1+(−2.03+3.99i)T+(−27.6−38.0i)T2 1 + (-2.03 + 3.99i)T + (-27.6 - 38.0i)T^{2}
53 1+(0.594−3.75i)T+(−50.4−16.3i)T2 1 + (0.594 - 3.75i)T + (-50.4 - 16.3i)T^{2}
59 1+(−4.27−1.38i)T+(47.7+34.6i)T2 1 + (-4.27 - 1.38i)T + (47.7 + 34.6i)T^{2}
61 1+(−3.48+4.79i)T+(−18.8−58.0i)T2 1 + (-3.48 + 4.79i)T + (-18.8 - 58.0i)T^{2}
67 1+(5.40+5.40i)T+67iT2 1 + (5.40 + 5.40i)T + 67iT^{2}
71 1+(7.19+5.23i)T+(21.9+67.5i)T2 1 + (7.19 + 5.23i)T + (21.9 + 67.5i)T^{2}
73 1+(−10.1+5.17i)T+(42.9−59.0i)T2 1 + (-10.1 + 5.17i)T + (42.9 - 59.0i)T^{2}
79 1+(4.69−3.40i)T+(24.4−75.1i)T2 1 + (4.69 - 3.40i)T + (24.4 - 75.1i)T^{2}
83 1+(−0.200−1.26i)T+(−78.9+25.6i)T2 1 + (-0.200 - 1.26i)T + (-78.9 + 25.6i)T^{2}
89 1+10.5iT−89T2 1 + 10.5iT - 89T^{2}
97 1+(6.92+1.09i)T+(92.2+29.9i)T2 1 + (6.92 + 1.09i)T + (92.2 + 29.9i)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

−10.71111200066158910704089888580, −9.983366987769231914223370041370, −8.648037579123344495804605874087, −7.60775323928613313186085906880, −6.96190852711155025746955514558, −6.12367837106725070632726034132, −5.19227317293486294358921569426, −4.69720482348279127004380132735, −2.24995317523736227293759851276, −0.47392733966921224581420842577, 1.14079428496128368764235087222, 3.33146406400879447505902248495, 4.37592607475896992128358148180, 5.24504009211110473549136970155, 5.99207349850656498794685808841, 7.25170011836935281093855677527, 8.426905060753745196655435710591, 9.728923492817438836433038286050, 10.27776280960866697252475220595, 10.88813549912848848968745449740

Graph of the ZZ-function along the critical line