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Properties

Label 2-220-44.19-c1-0-4
Degree 22
Conductor 220220
Sign 0.998+0.0519i0.998 + 0.0519i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
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  + (−0.117 − 1.40i)2-s + (−3.21 + 1.04i)3-s + (−1.97 + 0.331i)4-s + (−0.809 − 0.587i)5-s + (1.85 + 4.40i)6-s + (0.161 − 0.498i)7-s + (0.699 + 2.74i)8-s + (6.82 − 4.95i)9-s + (−0.733 + 1.20i)10-s + (2.73 + 1.87i)11-s + (5.99 − 3.12i)12-s + (2.22 + 3.06i)13-s + (−0.720 − 0.169i)14-s + (3.21 + 1.04i)15-s + (3.77 − 1.30i)16-s + (−1.63 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.0832 − 0.996i)2-s + (−1.85 + 0.603i)3-s + (−0.986 + 0.165i)4-s + (−0.361 − 0.262i)5-s + (0.755 + 1.79i)6-s + (0.0611 − 0.188i)7-s + (0.247 + 0.968i)8-s + (2.27 − 1.65i)9-s + (−0.231 + 0.382i)10-s + (0.824 + 0.566i)11-s + (1.73 − 0.902i)12-s + (0.617 + 0.850i)13-s + (−0.192 − 0.0452i)14-s + (0.830 + 0.269i)15-s + (0.944 − 0.327i)16-s + (−0.396 + 0.546i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.998+0.0519i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.998+0.0519i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.998+0.0519i0.998 + 0.0519i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(151,⋅)\chi_{220} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.998+0.0519i)(2,\ 220,\ (\ :1/2),\ 0.998 + 0.0519i)

Particular Values

L(1)L(1) ≈\approx 0.545584−0.0141696i0.545584 - 0.0141696i
L(12)L(\frac12) ≈\approx 0.545584−0.0141696i0.545584 - 0.0141696i
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+(0.117+1.40i)T 1 + (0.117 + 1.40i)T
5 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
11 1+(−2.73−1.87i)T 1 + (-2.73 - 1.87i)T
good3 1+(3.21−1.04i)T+(2.42−1.76i)T2 1 + (3.21 - 1.04i)T + (2.42 - 1.76i)T^{2}
7 1+(−0.161+0.498i)T+(−5.66−4.11i)T2 1 + (-0.161 + 0.498i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.22−3.06i)T+(−4.01+12.3i)T2 1 + (-2.22 - 3.06i)T + (-4.01 + 12.3i)T^{2}
17 1+(1.63−2.25i)T+(−5.25−16.1i)T2 1 + (1.63 - 2.25i)T + (-5.25 - 16.1i)T^{2}
19 1+(−0.531−1.63i)T+(−15.3+11.1i)T2 1 + (-0.531 - 1.63i)T + (-15.3 + 11.1i)T^{2}
23 1+4.45iT−23T2 1 + 4.45iT - 23T^{2}
29 1+(−0.474−0.154i)T+(23.4+17.0i)T2 1 + (-0.474 - 0.154i)T + (23.4 + 17.0i)T^{2}
31 1+(−2.66−3.67i)T+(−9.57+29.4i)T2 1 + (-2.66 - 3.67i)T + (-9.57 + 29.4i)T^{2}
37 1+(2.02−6.22i)T+(−29.9−21.7i)T2 1 + (2.02 - 6.22i)T + (-29.9 - 21.7i)T^{2}
41 1+(−3.86+1.25i)T+(33.1−24.0i)T2 1 + (-3.86 + 1.25i)T + (33.1 - 24.0i)T^{2}
43 1−9.72T+43T2 1 - 9.72T + 43T^{2}
47 1+(−6.44+2.09i)T+(38.0−27.6i)T2 1 + (-6.44 + 2.09i)T + (38.0 - 27.6i)T^{2}
53 1+(0.379−0.275i)T+(16.3−50.4i)T2 1 + (0.379 - 0.275i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.529−0.171i)T+(47.7+34.6i)T2 1 + (-0.529 - 0.171i)T + (47.7 + 34.6i)T^{2}
61 1+(5.95−8.19i)T+(−18.8−58.0i)T2 1 + (5.95 - 8.19i)T + (-18.8 - 58.0i)T^{2}
67 1+1.46iT−67T2 1 + 1.46iT - 67T^{2}
71 1+(−0.730+1.00i)T+(−21.9−67.5i)T2 1 + (-0.730 + 1.00i)T + (-21.9 - 67.5i)T^{2}
73 1+(−11.0−3.59i)T+(59.0+42.9i)T2 1 + (-11.0 - 3.59i)T + (59.0 + 42.9i)T^{2}
79 1+(9.67−7.03i)T+(24.4−75.1i)T2 1 + (9.67 - 7.03i)T + (24.4 - 75.1i)T^{2}
83 1+(3.87+2.81i)T+(25.6+78.9i)T2 1 + (3.87 + 2.81i)T + (25.6 + 78.9i)T^{2}
89 1+8.48T+89T2 1 + 8.48T + 89T^{2}
97 1+(6.27−4.55i)T+(29.9−92.2i)T2 1 + (6.27 - 4.55i)T + (29.9 - 92.2i)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

−12.20345696734604545934112658212, −11.24784986971566703790743493647, −10.66474653197776598266150460673, −9.742302930679247548299818972177, −8.765778317532832774133368689802, −6.95876354563406317205406832482, −5.83917520223754224398247852484, −4.47284413233946156202622883800, −4.07479864383826568586626514992, −1.21566058457944414330949118946, 0.76432794432755225004382625438, 4.15526623749758918471848145352, 5.44987397624670148384597903742, 6.09034848005571224108906793457, 7.03091236259624572570681691002, 7.85305637370945157706109229257, 9.294422306003187781506311968479, 10.62811121830001015251533910620, 11.33990253877818077436958223463, 12.27381040993286656814231793140

Graph of the ZZ-function along the critical line