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Properties

Label 2-60-20.19-c2-0-3
Degree 22
Conductor 6060
Sign 0.0450−0.998i0.0450 - 0.998i
Analytic cond. 1.634881.63488
Root an. cond. 1.278621.27862
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  + (0.656 + 1.88i)2-s + 1.73·3-s + (−3.13 + 2.48i)4-s + (−3.27 + 3.77i)5-s + (1.13 + 3.27i)6-s + 9.55·7-s + (−6.74 − 4.29i)8-s + 2.99·9-s + (−9.28 − 3.70i)10-s − 9.92i·11-s + (−5.43 + 4.29i)12-s − 7.55i·13-s + (6.27 + 18.0i)14-s + (−5.67 + 6.54i)15-s + (3.68 − 15.5i)16-s + 17.1i·17-s + ⋯
L(s)  = 1  + (0.328 + 0.944i)2-s + 0.577·3-s + (−0.784 + 0.620i)4-s + (−0.654 + 0.755i)5-s + (0.189 + 0.545i)6-s + 1.36·7-s + (−0.843 − 0.537i)8-s + 0.333·9-s + (−0.928 − 0.370i)10-s − 0.902i·11-s + (−0.452 + 0.358i)12-s − 0.581i·13-s + (0.448 + 1.28i)14-s + (−0.378 + 0.436i)15-s + (0.230 − 0.973i)16-s + 1.01i·17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.0450−0.998i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1) L(s)=((0.0450−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.0450−0.998i0.0450 - 0.998i
Analytic conductor: 1.634881.63488
Root analytic conductor: 1.278621.27862
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ60(19,⋅)\chi_{60} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1), 0.0450−0.998i)(2,\ 60,\ (\ :1),\ 0.0450 - 0.998i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.07195+1.02473i1.07195 + 1.02473i
L(12)L(\frac12) ≈\approx 1.07195+1.02473i1.07195 + 1.02473i
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+(−0.656−1.88i)T 1 + (-0.656 - 1.88i)T
3 1−1.73T 1 - 1.73T
5 1+(3.27−3.77i)T 1 + (3.27 - 3.77i)T
good7 1−9.55T+49T2 1 - 9.55T + 49T^{2}
11 1+9.92iT−121T2 1 + 9.92iT - 121T^{2}
13 1+7.55iT−169T2 1 + 7.55iT - 169T^{2}
17 1−17.1iT−289T2 1 - 17.1iT - 289T^{2}
19 1+26.1iT−361T2 1 + 26.1iT - 361T^{2}
23 1+1.67T+529T2 1 + 1.67T + 529T^{2}
29 1+0.350T+841T2 1 + 0.350T + 841T^{2}
31 1−46.0iT−961T2 1 - 46.0iT - 961T^{2}
37 1−22.6iT−1.36e3T2 1 - 22.6iT - 1.36e3T^{2}
41 1+77.2T+1.68e3T2 1 + 77.2T + 1.68e3T^{2}
43 1+41.7T+1.84e3T2 1 + 41.7T + 1.84e3T^{2}
47 1−14.0T+2.20e3T2 1 - 14.0T + 2.20e3T^{2}
53 1−22.6iT−2.80e3T2 1 - 22.6iT - 2.80e3T^{2}
59 1+94.7iT−3.48e3T2 1 + 94.7iT - 3.48e3T^{2}
61 1−38T+3.72e3T2 1 - 38T + 3.72e3T^{2}
67 1+29.8T+4.48e3T2 1 + 29.8T + 4.48e3T^{2}
71 1+7.19iT−5.04e3T2 1 + 7.19iT - 5.04e3T^{2}
73 1−34.3iT−5.32e3T2 1 - 34.3iT - 5.32e3T^{2}
79 1+46.0iT−6.24e3T2 1 + 46.0iT - 6.24e3T^{2}
83 1−24.1T+6.88e3T2 1 - 24.1T + 6.88e3T^{2}
89 1−100.T+7.92e3T2 1 - 100.T + 7.92e3T^{2}
97 1−131.iT−9.40e3T2 1 - 131. iT - 9.40e3T^{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

−15.03719153622697507577899352432, −14.30524887099817242193542668693, −13.32921610242223139309509579713, −11.80537173687935362383330555568, −10.60724053252316920222639707422, −8.596739074861972747520513027468, −8.009684354705356939162922559445, −6.72641944700508647733971289322, −4.94337561389568052467302108314, −3.37123330768847173465987819710, 1.78676721456718306322720195618, 4.05306091365102585056042690753, 5.06245046059594834524209294761, 7.70241667998069432171540694033, 8.771552468125978852607259120999, 9.967447050929617424449558024446, 11.51754396703250652407584911203, 12.11829963712505594469597310578, 13.40558127287845532450303828991, 14.46850310442382004915156459574

Graph of the ZZ-function along the critical line