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Properties

Label 2-300-5.4-c3-0-1
Degree 22
Conductor 300300
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 17.700517.7005
Root an. cond. 4.207204.20720
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 28i·7-s − 9·9-s − 24·11-s + 70i·13-s + 102i·17-s − 20·19-s + 84·21-s + 72i·23-s − 27i·27-s − 306·29-s − 136·31-s − 72i·33-s − 214i·37-s − 210·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 0.657·11-s + 1.49i·13-s + 1.45i·17-s − 0.241·19-s + 0.872·21-s + 0.652i·23-s − 0.192i·27-s − 1.95·29-s − 0.787·31-s − 0.379i·33-s − 0.950i·37-s − 0.862·39-s + ⋯

Functional equation

Λ(s)=(300s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(300s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 300300    =    22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 17.700517.7005
Root analytic conductor: 4.207204.20720
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ300(49,⋅)\chi_{300} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 300, ( :3/2), −0.894−0.447i)(2,\ 300,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 0.61681541410.6168154141
L(12)L(\frac12) ≈\approx 0.61681541410.6168154141
L(52)L(\frac{5}{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
3 1−3iT 1 - 3iT
5 1 1
good7 1+28iT−343T2 1 + 28iT - 343T^{2}
11 1+24T+1.33e3T2 1 + 24T + 1.33e3T^{2}
13 1−70iT−2.19e3T2 1 - 70iT - 2.19e3T^{2}
17 1−102iT−4.91e3T2 1 - 102iT - 4.91e3T^{2}
19 1+20T+6.85e3T2 1 + 20T + 6.85e3T^{2}
23 1−72iT−1.21e4T2 1 - 72iT - 1.21e4T^{2}
29 1+306T+2.43e4T2 1 + 306T + 2.43e4T^{2}
31 1+136T+2.97e4T2 1 + 136T + 2.97e4T^{2}
37 1+214iT−5.06e4T2 1 + 214iT - 5.06e4T^{2}
41 1+150T+6.89e4T2 1 + 150T + 6.89e4T^{2}
43 1−292iT−7.95e4T2 1 - 292iT - 7.95e4T^{2}
47 1+72iT−1.03e5T2 1 + 72iT - 1.03e5T^{2}
53 1−414iT−1.48e5T2 1 - 414iT - 1.48e5T^{2}
59 1−744T+2.05e5T2 1 - 744T + 2.05e5T^{2}
61 1+418T+2.26e5T2 1 + 418T + 2.26e5T^{2}
67 1−188iT−3.00e5T2 1 - 188iT - 3.00e5T^{2}
71 1−480T+3.57e5T2 1 - 480T + 3.57e5T^{2}
73 1+434iT−3.89e5T2 1 + 434iT - 3.89e5T^{2}
79 1+1.35e3T+4.93e5T2 1 + 1.35e3T + 4.93e5T^{2}
83 1−612iT−5.71e5T2 1 - 612iT - 5.71e5T^{2}
89 1−30T+7.04e5T2 1 - 30T + 7.04e5T^{2}
97 1+286iT−9.12e5T2 1 + 286iT - 9.12e5T^{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

−11.34064297322135270274776709491, −10.79881034349280661534942936398, −9.931593722838556417466438636177, −9.014578247177705852645739610895, −7.79503360970758461673710478982, −6.94549049681046813404907980585, −5.68441340119835387532225692903, −4.29532029313424573080235283742, −3.70522203594383587188523298040, −1.73700742106261002077288777173, 0.21581774694381423481913778020, 2.19196243165657908107263875991, 3.12361369486874690740920191813, 5.21092384867302336260013302716, 5.69885639492656934443017549807, 7.07107147013258066738460917740, 8.067908766606554171168212463315, 8.870127374877108429017804567616, 9.904395766119424214776347652034, 11.08878124790016621636380164670

Graph of the ZZ-function along the critical line