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Properties

Label 2-60-20.19-c2-0-2
Degree 22
Conductor 6060
Sign 0.648−0.761i0.648 - 0.761i
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  + (−1.52 + 1.29i)2-s + 1.73·3-s + (0.637 − 3.94i)4-s + (4.27 + 2.59i)5-s + (−2.63 + 2.24i)6-s + 0.837·7-s + (4.14 + 6.83i)8-s + 2.99·9-s + (−9.87 + 1.59i)10-s + 15.7i·11-s + (1.10 − 6.83i)12-s − 5.18i·13-s + (−1.27 + 1.08i)14-s + (7.40 + 4.49i)15-s + (−15.1 − 5.03i)16-s − 27.3i·17-s + ⋯
L(s)  = 1  + (−0.761 + 0.648i)2-s + 0.577·3-s + (0.159 − 0.987i)4-s + (0.854 + 0.518i)5-s + (−0.439 + 0.374i)6-s + 0.119·7-s + (0.518 + 0.854i)8-s + 0.333·9-s + (−0.987 + 0.159i)10-s + 1.43i·11-s + (0.0920 − 0.569i)12-s − 0.398i·13-s + (−0.0910 + 0.0775i)14-s + (0.493 + 0.299i)15-s + (−0.949 − 0.314i)16-s − 1.60i·17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.648−0.761i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1) L(s)=((0.648−0.761i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.648−0.761i0.648 - 0.761i
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), 0.648−0.761i)(2,\ 60,\ (\ :1),\ 0.648 - 0.761i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.999888+0.461884i0.999888 + 0.461884i
L(12)L(\frac12) ≈\approx 0.999888+0.461884i0.999888 + 0.461884i
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+(1.52−1.29i)T 1 + (1.52 - 1.29i)T
3 1−1.73T 1 - 1.73T
5 1+(−4.27−2.59i)T 1 + (-4.27 - 2.59i)T
good7 1−0.837T+49T2 1 - 0.837T + 49T^{2}
11 1−15.7iT−121T2 1 - 15.7iT - 121T^{2}
13 1+5.18iT−169T2 1 + 5.18iT - 169T^{2}
17 1+27.3iT−289T2 1 + 27.3iT - 289T^{2}
19 1+17.9iT−361T2 1 + 17.9iT - 361T^{2}
23 1+19.1T+529T2 1 + 19.1T + 529T^{2}
29 1+45.6T+841T2 1 + 45.6T + 841T^{2}
31 1+13.6iT−961T2 1 + 13.6iT - 961T^{2}
37 1−15.5iT−1.36e3T2 1 - 15.5iT - 1.36e3T^{2}
41 1−13.2T+1.68e3T2 1 - 13.2T + 1.68e3T^{2}
43 1−27.9T+1.84e3T2 1 - 27.9T + 1.84e3T^{2}
47 1+55.6T+2.20e3T2 1 + 55.6T + 2.20e3T^{2}
53 1−15.5iT−2.80e3T2 1 - 15.5iT - 2.80e3T^{2}
59 1+87.6iT−3.48e3T2 1 + 87.6iT - 3.48e3T^{2}
61 1−38T+3.72e3T2 1 - 38T + 3.72e3T^{2}
67 1−92.2T+4.48e3T2 1 - 92.2T + 4.48e3T^{2}
71 1−130.iT−5.04e3T2 1 - 130. iT - 5.04e3T^{2}
73 1+54.7iT−5.32e3T2 1 + 54.7iT - 5.32e3T^{2}
79 1−13.6iT−6.24e3T2 1 - 13.6iT - 6.24e3T^{2}
83 1−59.0T+6.88e3T2 1 - 59.0T + 6.88e3T^{2}
89 1−39.8T+7.92e3T2 1 - 39.8T + 7.92e3T^{2}
97 1−168.iT−9.40e3T2 1 - 168. 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

−14.99340960891530776773131741442, −14.22527848756251541860591268488, −13.12520099753244118854640497809, −11.25901092296721083301447069403, −9.844720747989565891884421879196, −9.377694814379083670399414741513, −7.69537566720917626542710762451, −6.76390497164608425220034557678, −5.09574977081002269468318713733, −2.24036874985020074972194857390, 1.77107311513187937728737987597, 3.72819777884412453162754378647, 6.06231484544028540828553644927, 8.040969287720339438543559879528, 8.838233453312031952792226007468, 9.948824424558109592484640858801, 11.04265271036653657177746224578, 12.50493862133201825103804860246, 13.39984026842115638375473390368, 14.46604111881252011939755318407

Graph of the ZZ-function along the critical line