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Properties

Label 2-220-44.19-c1-0-10
Degree 22
Conductor 220220
Sign 0.785+0.619i0.785 + 0.619i
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  + (−1.21 − 0.727i)2-s + (0.919 − 0.298i)3-s + (0.941 + 1.76i)4-s + (0.809 + 0.587i)5-s + (−1.33 − 0.306i)6-s + (0.819 − 2.52i)7-s + (0.141 − 2.82i)8-s + (−1.67 + 1.21i)9-s + (−0.553 − 1.30i)10-s + (3.31 − 0.0962i)11-s + (1.39 + 1.34i)12-s + (3.18 + 4.38i)13-s + (−2.82 + 2.46i)14-s + (0.919 + 0.298i)15-s + (−2.22 + 3.32i)16-s + (2.85 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.857 − 0.514i)2-s + (0.530 − 0.172i)3-s + (0.470 + 0.882i)4-s + (0.361 + 0.262i)5-s + (−0.543 − 0.125i)6-s + (0.309 − 0.953i)7-s + (0.0501 − 0.998i)8-s + (−0.557 + 0.404i)9-s + (−0.175 − 0.411i)10-s + (0.999 − 0.0290i)11-s + (0.401 + 0.387i)12-s + (0.884 + 1.21i)13-s + (−0.756 + 0.658i)14-s + (0.237 + 0.0771i)15-s + (−0.556 + 0.830i)16-s + (0.692 − 0.952i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.785+0.619i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.785+0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.785+0.619i0.785 + 0.619i
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), 0.785+0.619i)(2,\ 220,\ (\ :1/2),\ 0.785 + 0.619i)

Particular Values

L(1)L(1) ≈\approx 1.02350−0.355125i1.02350 - 0.355125i
L(12)L(\frac12) ≈\approx 1.02350−0.355125i1.02350 - 0.355125i
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+(1.21+0.727i)T 1 + (1.21 + 0.727i)T
5 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1+(−3.31+0.0962i)T 1 + (-3.31 + 0.0962i)T
good3 1+(−0.919+0.298i)T+(2.42−1.76i)T2 1 + (-0.919 + 0.298i)T + (2.42 - 1.76i)T^{2}
7 1+(−0.819+2.52i)T+(−5.66−4.11i)T2 1 + (-0.819 + 2.52i)T + (-5.66 - 4.11i)T^{2}
13 1+(−3.18−4.38i)T+(−4.01+12.3i)T2 1 + (-3.18 - 4.38i)T + (-4.01 + 12.3i)T^{2}
17 1+(−2.85+3.92i)T+(−5.25−16.1i)T2 1 + (-2.85 + 3.92i)T + (-5.25 - 16.1i)T^{2}
19 1+(0.860+2.64i)T+(−15.3+11.1i)T2 1 + (0.860 + 2.64i)T + (-15.3 + 11.1i)T^{2}
23 1+6.73iT−23T2 1 + 6.73iT - 23T^{2}
29 1+(5.27+1.71i)T+(23.4+17.0i)T2 1 + (5.27 + 1.71i)T + (23.4 + 17.0i)T^{2}
31 1+(−2.76−3.81i)T+(−9.57+29.4i)T2 1 + (-2.76 - 3.81i)T + (-9.57 + 29.4i)T^{2}
37 1+(2.07−6.38i)T+(−29.9−21.7i)T2 1 + (2.07 - 6.38i)T + (-29.9 - 21.7i)T^{2}
41 1+(5.13−1.66i)T+(33.1−24.0i)T2 1 + (5.13 - 1.66i)T + (33.1 - 24.0i)T^{2}
43 1−5.61T+43T2 1 - 5.61T + 43T^{2}
47 1+(11.2−3.66i)T+(38.0−27.6i)T2 1 + (11.2 - 3.66i)T + (38.0 - 27.6i)T^{2}
53 1+(−4.06+2.95i)T+(16.3−50.4i)T2 1 + (-4.06 + 2.95i)T + (16.3 - 50.4i)T^{2}
59 1+(3.91+1.27i)T+(47.7+34.6i)T2 1 + (3.91 + 1.27i)T + (47.7 + 34.6i)T^{2}
61 1+(4.69−6.46i)T+(−18.8−58.0i)T2 1 + (4.69 - 6.46i)T + (-18.8 - 58.0i)T^{2}
67 1+1.99iT−67T2 1 + 1.99iT - 67T^{2}
71 1+(7.82−10.7i)T+(−21.9−67.5i)T2 1 + (7.82 - 10.7i)T + (-21.9 - 67.5i)T^{2}
73 1+(10.0+3.26i)T+(59.0+42.9i)T2 1 + (10.0 + 3.26i)T + (59.0 + 42.9i)T^{2}
79 1+(9.25−6.72i)T+(24.4−75.1i)T2 1 + (9.25 - 6.72i)T + (24.4 - 75.1i)T^{2}
83 1+(7.51+5.45i)T+(25.6+78.9i)T2 1 + (7.51 + 5.45i)T + (25.6 + 78.9i)T^{2}
89 1−14.5T+89T2 1 - 14.5T + 89T^{2}
97 1+(−2.55+1.85i)T+(29.9−92.2i)T2 1 + (-2.55 + 1.85i)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

−11.77630045264357452672409503736, −11.20451168176233635852199506280, −10.22161078704056332850317185354, −9.156550160402639284822714148223, −8.469596878807816901635015270290, −7.29199475759097600807741301574, −6.47584689261077473610732859057, −4.34396262172962019733651815047, −3.01546319558023640236224805950, −1.50842593829999927508949326669, 1.67768487296800924983497055325, 3.46752358364003216049884494716, 5.64760165041214599391904731928, 6.01258246756491939269692195448, 7.74686810386392356856377071795, 8.592443811454467566825596151921, 9.168087121596435532768775112872, 10.13325328496421396659319767848, 11.30492544636048388654637106616, 12.20955779200205449296141618888

Graph of the ZZ-function along the critical line