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Properties

Label 2-220-11.2-c2-0-3
Degree 22
Conductor 220220
Sign 0.976−0.217i0.976 - 0.217i
Analytic cond. 5.994565.99456
Root an. cond. 2.448382.44838
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.86 + 2.80i)3-s + (−0.690 − 2.12i)5-s + (5.14 − 7.08i)7-s + (4.27 − 13.1i)9-s + (−4.15 + 10.1i)11-s + (−6.21 − 2.02i)13-s + (8.64 + 6.28i)15-s + (30.3 − 9.85i)17-s + (15.4 + 21.3i)19-s + 41.8i·21-s + 41.5·23-s + (−4.04 + 2.93i)25-s + (7.16 + 22.0i)27-s + (10.2 − 14.1i)29-s + (−4.59 + 14.1i)31-s + ⋯
L(s)  = 1  + (−1.28 + 0.936i)3-s + (−0.138 − 0.425i)5-s + (0.735 − 1.01i)7-s + (0.475 − 1.46i)9-s + (−0.377 + 0.926i)11-s + (−0.478 − 0.155i)13-s + (0.576 + 0.418i)15-s + (1.78 − 0.579i)17-s + (0.814 + 1.12i)19-s + 1.99i·21-s + 1.80·23-s + (−0.161 + 0.117i)25-s + (0.265 + 0.816i)27-s + (0.355 − 0.488i)29-s + (−0.148 + 0.456i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.976−0.217i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((0.976−0.217i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.976−0.217i0.976 - 0.217i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(101,⋅)\chi_{220} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), 0.976−0.217i)(2,\ 220,\ (\ :1),\ 0.976 - 0.217i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.04639+0.115174i1.04639 + 0.115174i
L(12)L(\frac12) ≈\approx 1.04639+0.115174i1.04639 + 0.115174i
L(2)L(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
5 1+(0.690+2.12i)T 1 + (0.690 + 2.12i)T
11 1+(4.15−10.1i)T 1 + (4.15 - 10.1i)T
good3 1+(3.86−2.80i)T+(2.78−8.55i)T2 1 + (3.86 - 2.80i)T + (2.78 - 8.55i)T^{2}
7 1+(−5.14+7.08i)T+(−15.1−46.6i)T2 1 + (-5.14 + 7.08i)T + (-15.1 - 46.6i)T^{2}
13 1+(6.21+2.02i)T+(136.+99.3i)T2 1 + (6.21 + 2.02i)T + (136. + 99.3i)T^{2}
17 1+(−30.3+9.85i)T+(233.−169.i)T2 1 + (-30.3 + 9.85i)T + (233. - 169. i)T^{2}
19 1+(−15.4−21.3i)T+(−111.+343.i)T2 1 + (-15.4 - 21.3i)T + (-111. + 343. i)T^{2}
23 1−41.5T+529T2 1 - 41.5T + 529T^{2}
29 1+(−10.2+14.1i)T+(−259.−799.i)T2 1 + (-10.2 + 14.1i)T + (-259. - 799. i)T^{2}
31 1+(4.59−14.1i)T+(−777.−564.i)T2 1 + (4.59 - 14.1i)T + (-777. - 564. i)T^{2}
37 1+(27.3+19.8i)T+(423.+1.30e3i)T2 1 + (27.3 + 19.8i)T + (423. + 1.30e3i)T^{2}
41 1+(−2.22−3.05i)T+(−519.+1.59e3i)T2 1 + (-2.22 - 3.05i)T + (-519. + 1.59e3i)T^{2}
43 1−23.8iT−1.84e3T2 1 - 23.8iT - 1.84e3T^{2}
47 1+(−20.2+14.7i)T+(682.−2.10e3i)T2 1 + (-20.2 + 14.7i)T + (682. - 2.10e3i)T^{2}
53 1+(−32.5+100.i)T+(−2.27e3−1.65e3i)T2 1 + (-32.5 + 100. i)T + (-2.27e3 - 1.65e3i)T^{2}
59 1+(49.2+35.8i)T+(1.07e3+3.31e3i)T2 1 + (49.2 + 35.8i)T + (1.07e3 + 3.31e3i)T^{2}
61 1+(−76.1+24.7i)T+(3.01e3−2.18e3i)T2 1 + (-76.1 + 24.7i)T + (3.01e3 - 2.18e3i)T^{2}
67 1−109.T+4.48e3T2 1 - 109.T + 4.48e3T^{2}
71 1+(10.4+32.2i)T+(−4.07e3+2.96e3i)T2 1 + (10.4 + 32.2i)T + (-4.07e3 + 2.96e3i)T^{2}
73 1+(23.7−32.6i)T+(−1.64e3−5.06e3i)T2 1 + (23.7 - 32.6i)T + (-1.64e3 - 5.06e3i)T^{2}
79 1+(−66.5−21.6i)T+(5.04e3+3.66e3i)T2 1 + (-66.5 - 21.6i)T + (5.04e3 + 3.66e3i)T^{2}
83 1+(42.8−13.9i)T+(5.57e3−4.04e3i)T2 1 + (42.8 - 13.9i)T + (5.57e3 - 4.04e3i)T^{2}
89 1+24.9T+7.92e3T2 1 + 24.9T + 7.92e3T^{2}
97 1+(39.7−122.i)T+(−7.61e3−5.53e3i)T2 1 + (39.7 - 122. i)T + (-7.61e3 - 5.53e3i)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

−11.95366024008989124390342084257, −11.08249377593587570047963483225, −10.16016762046936389002123462122, −9.676689244531214083443559310672, −7.927389108728491040233635248346, −7.05163233109855889502658889863, −5.28583132092568521448431126958, −5.00313236478724114101438289257, −3.71597062003659372104514192025, −0.956457154888133134243995279363, 1.07605229318577362977115656604, 2.87116620930587495813513111281, 5.19336268760880421288537122566, 5.64209396336760779783072662277, 6.91041208190727031768849528814, 7.76184429581966907724568616035, 8.945833162779363040016513084027, 10.50568298852932370527583487736, 11.31059047589562566947225174744, 11.92546601200809301971757951529

Graph of the ZZ-function along the critical line