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Properties

Label 2-300-15.14-c2-0-5
Degree 22
Conductor 300300
Sign 0.807−0.590i0.807 - 0.590i
Analytic cond. 8.174408.17440
Root an. cond. 2.859092.85909
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  + (2.95 − 0.5i)3-s + 8i·7-s + (8.5 − 2.95i)9-s + 17.7i·11-s − 2i·13-s + 17.7·17-s − 11·19-s + (4 + 23.6i)21-s + 35.4·23-s + (23.6 − 13i)27-s − 35.4i·29-s − 46·31-s + (8.87 + 52.5i)33-s − 16i·37-s + (−1 − 5.91i)39-s + ⋯
L(s)  = 1  + (0.986 − 0.166i)3-s + 1.14i·7-s + (0.944 − 0.328i)9-s + 1.61i·11-s − 0.153i·13-s + 1.04·17-s − 0.578·19-s + (0.190 + 1.12i)21-s + 1.54·23-s + (0.876 − 0.481i)27-s − 1.22i·29-s − 1.48·31-s + (0.268 + 1.59i)33-s − 0.432i·37-s + (−0.0256 − 0.151i)39-s + ⋯

Functional equation

Λ(s)=(300s/2 ΓC(s) L(s)=((0.807−0.590i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(300s/2 ΓC(s+1) L(s)=((0.807−0.590i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 300300    =    22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}
Sign: 0.807−0.590i0.807 - 0.590i
Analytic conductor: 8.174408.17440
Root analytic conductor: 2.859092.85909
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ300(149,⋅)\chi_{300} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 300, ( :1), 0.807−0.590i)(2,\ 300,\ (\ :1),\ 0.807 - 0.590i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.19162+0.715467i2.19162 + 0.715467i
L(12)L(\frac12) ≈\approx 2.19162+0.715467i2.19162 + 0.715467i
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
3 1+(−2.95+0.5i)T 1 + (-2.95 + 0.5i)T
5 1 1
good7 1−8iT−49T2 1 - 8iT - 49T^{2}
11 1−17.7iT−121T2 1 - 17.7iT - 121T^{2}
13 1+2iT−169T2 1 + 2iT - 169T^{2}
17 1−17.7T+289T2 1 - 17.7T + 289T^{2}
19 1+11T+361T2 1 + 11T + 361T^{2}
23 1−35.4T+529T2 1 - 35.4T + 529T^{2}
29 1+35.4iT−841T2 1 + 35.4iT - 841T^{2}
31 1+46T+961T2 1 + 46T + 961T^{2}
37 1+16iT−1.36e3T2 1 + 16iT - 1.36e3T^{2}
41 1−53.2iT−1.68e3T2 1 - 53.2iT - 1.68e3T^{2}
43 1+62iT−1.84e3T2 1 + 62iT - 1.84e3T^{2}
47 1−35.4T+2.20e3T2 1 - 35.4T + 2.20e3T^{2}
53 1+35.4T+2.80e3T2 1 + 35.4T + 2.80e3T^{2}
59 1−70.9iT−3.48e3T2 1 - 70.9iT - 3.48e3T^{2}
61 1+16T+3.72e3T2 1 + 16T + 3.72e3T^{2}
67 1−113iT−4.48e3T2 1 - 113iT - 4.48e3T^{2}
71 1+106.iT−5.04e3T2 1 + 106. iT - 5.04e3T^{2}
73 1+101iT−5.32e3T2 1 + 101iT - 5.32e3T^{2}
79 1+68T+6.24e3T2 1 + 68T + 6.24e3T^{2}
83 1+17.7T+6.88e3T2 1 + 17.7T + 6.88e3T^{2}
89 1+53.2iT−7.92e3T2 1 + 53.2iT - 7.92e3T^{2}
97 1+22iT−9.40e3T2 1 + 22iT - 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

−11.92297636633297638917520635762, −10.46553843191443300928557737252, −9.495068947522083547376367144686, −8.905890747446024166977293500952, −7.77840162007580419900345731076, −6.99626340442483195494221904805, −5.58088591713682884143870267690, −4.33569133529852055960664991271, −2.89701602767129349658174197167, −1.83328191531635504418544631323, 1.15344968064700163055068012612, 3.09538553677403797475582856573, 3.85639682646753472871634804551, 5.26040036256378964311798677800, 6.77152207805068020533276548506, 7.66500390149678367963005145053, 8.598538582553715181023395968327, 9.410529747760460790978163433070, 10.61757081764020898536996632337, 11.04458868014183475795383365079

Graph of the ZZ-function along the critical line