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Properties

Label 2-220-11.2-c2-0-5
Degree 22
Conductor 220220
Sign 0.653+0.757i0.653 + 0.757i
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  + (4.71 − 3.42i)3-s + (0.690 + 2.12i)5-s + (−2.62 + 3.61i)7-s + (7.70 − 23.7i)9-s + (10.9 − 1.15i)11-s + (9.85 + 3.20i)13-s + (10.5 + 7.65i)15-s + (0.409 − 0.133i)17-s + (−15.0 − 20.6i)19-s + 26.0i·21-s − 21.0·23-s + (−4.04 + 2.93i)25-s + (−28.6 − 88.2i)27-s + (−31.7 + 43.7i)29-s + (−3.90 + 12.0i)31-s + ⋯
L(s)  = 1  + (1.57 − 1.14i)3-s + (0.138 + 0.425i)5-s + (−0.375 + 0.516i)7-s + (0.855 − 2.63i)9-s + (0.994 − 0.105i)11-s + (0.757 + 0.246i)13-s + (0.702 + 0.510i)15-s + (0.0241 − 0.00783i)17-s + (−0.790 − 1.08i)19-s + 1.24i·21-s − 0.915·23-s + (−0.161 + 0.117i)25-s + (−1.06 − 3.26i)27-s + (−1.09 + 1.50i)29-s + (−0.126 + 0.388i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.653+0.757i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((0.653+0.757i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.653+0.757i0.653 + 0.757i
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), 0.653+0.757i)(2,\ 220,\ (\ :1),\ 0.653 + 0.757i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.34571−1.07437i2.34571 - 1.07437i
L(12)L(\frac12) ≈\approx 2.34571−1.07437i2.34571 - 1.07437i
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+(−10.9+1.15i)T 1 + (-10.9 + 1.15i)T
good3 1+(−4.71+3.42i)T+(2.78−8.55i)T2 1 + (-4.71 + 3.42i)T + (2.78 - 8.55i)T^{2}
7 1+(2.62−3.61i)T+(−15.1−46.6i)T2 1 + (2.62 - 3.61i)T + (-15.1 - 46.6i)T^{2}
13 1+(−9.85−3.20i)T+(136.+99.3i)T2 1 + (-9.85 - 3.20i)T + (136. + 99.3i)T^{2}
17 1+(−0.409+0.133i)T+(233.−169.i)T2 1 + (-0.409 + 0.133i)T + (233. - 169. i)T^{2}
19 1+(15.0+20.6i)T+(−111.+343.i)T2 1 + (15.0 + 20.6i)T + (-111. + 343. i)T^{2}
23 1+21.0T+529T2 1 + 21.0T + 529T^{2}
29 1+(31.7−43.7i)T+(−259.−799.i)T2 1 + (31.7 - 43.7i)T + (-259. - 799. i)T^{2}
31 1+(3.90−12.0i)T+(−777.−564.i)T2 1 + (3.90 - 12.0i)T + (-777. - 564. i)T^{2}
37 1+(3.37+2.45i)T+(423.+1.30e3i)T2 1 + (3.37 + 2.45i)T + (423. + 1.30e3i)T^{2}
41 1+(−24.6−33.9i)T+(−519.+1.59e3i)T2 1 + (-24.6 - 33.9i)T + (-519. + 1.59e3i)T^{2}
43 1−84.6iT−1.84e3T2 1 - 84.6iT - 1.84e3T^{2}
47 1+(−34.2+24.9i)T+(682.−2.10e3i)T2 1 + (-34.2 + 24.9i)T + (682. - 2.10e3i)T^{2}
53 1+(−13.5+41.8i)T+(−2.27e3−1.65e3i)T2 1 + (-13.5 + 41.8i)T + (-2.27e3 - 1.65e3i)T^{2}
59 1+(53.6+38.9i)T+(1.07e3+3.31e3i)T2 1 + (53.6 + 38.9i)T + (1.07e3 + 3.31e3i)T^{2}
61 1+(39.7−12.9i)T+(3.01e3−2.18e3i)T2 1 + (39.7 - 12.9i)T + (3.01e3 - 2.18e3i)T^{2}
67 1−41.3T+4.48e3T2 1 - 41.3T + 4.48e3T^{2}
71 1+(−1.24−3.84i)T+(−4.07e3+2.96e3i)T2 1 + (-1.24 - 3.84i)T + (-4.07e3 + 2.96e3i)T^{2}
73 1+(−9.80+13.4i)T+(−1.64e3−5.06e3i)T2 1 + (-9.80 + 13.4i)T + (-1.64e3 - 5.06e3i)T^{2}
79 1+(−49.2−15.9i)T+(5.04e3+3.66e3i)T2 1 + (-49.2 - 15.9i)T + (5.04e3 + 3.66e3i)T^{2}
83 1+(−60.4+19.6i)T+(5.57e3−4.04e3i)T2 1 + (-60.4 + 19.6i)T + (5.57e3 - 4.04e3i)T^{2}
89 1+30.4T+7.92e3T2 1 + 30.4T + 7.92e3T^{2}
97 1+(52.9−163.i)T+(−7.61e3−5.53e3i)T2 1 + (52.9 - 163. 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

−12.26064765215225972403097834557, −11.09392587044250296080438151884, −9.426909680054938921268925226513, −8.988181385177101902636841138568, −8.002534042273057667805975089658, −6.84564717630924764202244128501, −6.25772485139886721749279125758, −3.86947417664274592853122111408, −2.81821486584447587077514974375, −1.57425100992281860418893236369, 2.04200917807691340003595277053, 3.80866903562551400872761308352, 4.08487932749044147863016583458, 5.85238784156827764058008380381, 7.54587049632882690213902995358, 8.467882360151508725264371433983, 9.252377502118124398257015151933, 10.01790776087061005413490089863, 10.82559724845409501878879305217, 12.35482935414957873774563604295

Graph of the ZZ-function along the critical line