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Properties

Label 2-300-15.14-c2-0-3
Degree 22
Conductor 300300
Sign 0.929−0.368i0.929 - 0.368i
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  + (−2.23 − 2i)3-s + 2i·7-s + (1.00 + 8.94i)9-s + 13.4i·11-s − 8i·13-s − 13.4·17-s + 34·19-s + (4 − 4.47i)21-s + 40.2·23-s + (15.6 − 22.0i)27-s + 40.2i·29-s + 14·31-s + (26.8 − 30.0i)33-s + 56i·37-s + (−16 + 17.8i)39-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)3-s + 0.285i·7-s + (0.111 + 0.993i)9-s + 1.21i·11-s − 0.615i·13-s − 0.789·17-s + 1.78·19-s + (0.190 − 0.212i)21-s + 1.74·23-s + (0.579 − 0.814i)27-s + 1.38i·29-s + 0.451·31-s + (0.813 − 0.909i)33-s + 1.51i·37-s + (−0.410 + 0.458i)39-s + ⋯

Functional equation

Λ(s)=(300s/2 ΓC(s) L(s)=((0.929−0.368i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(300s/2 ΓC(s+1) L(s)=((0.929−0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 300300    =    22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}
Sign: 0.929−0.368i0.929 - 0.368i
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), 0.929−0.368i)(2,\ 300,\ (\ :1),\ 0.929 - 0.368i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.18112+0.225574i1.18112 + 0.225574i
L(12)L(\frac12) ≈\approx 1.18112+0.225574i1.18112 + 0.225574i
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+(2.23+2i)T 1 + (2.23 + 2i)T
5 1 1
good7 1−2iT−49T2 1 - 2iT - 49T^{2}
11 1−13.4iT−121T2 1 - 13.4iT - 121T^{2}
13 1+8iT−169T2 1 + 8iT - 169T^{2}
17 1+13.4T+289T2 1 + 13.4T + 289T^{2}
19 1−34T+361T2 1 - 34T + 361T^{2}
23 1−40.2T+529T2 1 - 40.2T + 529T^{2}
29 1−40.2iT−841T2 1 - 40.2iT - 841T^{2}
31 1−14T+961T2 1 - 14T + 961T^{2}
37 1−56iT−1.36e3T2 1 - 56iT - 1.36e3T^{2}
41 1+26.8iT−1.68e3T2 1 + 26.8iT - 1.68e3T^{2}
43 1+8iT−1.84e3T2 1 + 8iT - 1.84e3T^{2}
47 1−40.2T+2.20e3T2 1 - 40.2T + 2.20e3T^{2}
53 1+40.2T+2.80e3T2 1 + 40.2T + 2.80e3T^{2}
59 1+13.4iT−3.48e3T2 1 + 13.4iT - 3.48e3T^{2}
61 1+46T+3.72e3T2 1 + 46T + 3.72e3T^{2}
67 1−32iT−4.48e3T2 1 - 32iT - 4.48e3T^{2}
71 1−53.6iT−5.04e3T2 1 - 53.6iT - 5.04e3T^{2}
73 1−106iT−5.32e3T2 1 - 106iT - 5.32e3T^{2}
79 1−22T+6.24e3T2 1 - 22T + 6.24e3T^{2}
83 1+120.T+6.88e3T2 1 + 120.T + 6.88e3T^{2}
89 1+107.iT−7.92e3T2 1 + 107. iT - 7.92e3T^{2}
97 1−122iT−9.40e3T2 1 - 122iT - 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

−11.67491287396899532945683869913, −10.78814719538746616487173932599, −9.816994786239068141701716547466, −8.687171284818910508052261474275, −7.38465657961656225191887835176, −6.88568984473372541655652726549, −5.49875699226628949783339057997, −4.76656131635516598791009424454, −2.83405182838099226883775213756, −1.25078637838298401631685047854, 0.77475133942346317145728847786, 3.13385976447391963677200149841, 4.33337270366909505230202698452, 5.42653055467533229033956953344, 6.38054458799849057792642677908, 7.47909025193132483199793278401, 8.927268389431399089464542199731, 9.545643525936812910738848792386, 10.78807501206847077592719275585, 11.28345877427665201339789754580

Graph of the ZZ-function along the critical line