↑

Properties

Label 2-464-116.115-c2-0-4
Degree 22
Conductor 464464
Sign 0.435−0.900i0.435 - 0.900i
Analytic cond. 12.643012.6430
Root an. cond. 3.555713.55571
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.54·3-s − 5.75·5-s − 7.46i·7-s + 3.55·9-s + 9.81·11-s − 24.4·13-s + 20.4·15-s − 31.6i·17-s − 10.6·19-s + 26.4i·21-s + 31.8i·23-s + 8.16·25-s + 19.2·27-s + (16.2 + 23.9i)29-s + 43.2·31-s + ⋯
L(s)  = 1  − 1.18·3-s − 1.15·5-s − 1.06i·7-s + 0.394·9-s + 0.892·11-s − 1.88·13-s + 1.36·15-s − 1.86i·17-s − 0.562·19-s + 1.25i·21-s + 1.38i·23-s + 0.326·25-s + 0.714·27-s + (0.561 + 0.827i)29-s + 1.39·31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.435−0.900i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1) L(s)=((0.435−0.900i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.435−0.900i0.435 - 0.900i
Analytic conductor: 12.643012.6430
Root analytic conductor: 3.555713.55571
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ464(463,⋅)\chi_{464} (463, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1), 0.435−0.900i)(2,\ 464,\ (\ :1),\ 0.435 - 0.900i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.38062651270.3806265127
L(12)L(\frac12) ≈\approx 0.38062651270.3806265127
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
29 1+(−16.2−23.9i)T 1 + (-16.2 - 23.9i)T
good3 1+3.54T+9T2 1 + 3.54T + 9T^{2}
5 1+5.75T+25T2 1 + 5.75T + 25T^{2}
7 1+7.46iT−49T2 1 + 7.46iT - 49T^{2}
11 1−9.81T+121T2 1 - 9.81T + 121T^{2}
13 1+24.4T+169T2 1 + 24.4T + 169T^{2}
17 1+31.6iT−289T2 1 + 31.6iT - 289T^{2}
19 1+10.6T+361T2 1 + 10.6T + 361T^{2}
23 1−31.8iT−529T2 1 - 31.8iT - 529T^{2}
31 1−43.2T+961T2 1 - 43.2T + 961T^{2}
37 1−31.4iT−1.36e3T2 1 - 31.4iT - 1.36e3T^{2}
41 1−58.5iT−1.68e3T2 1 - 58.5iT - 1.68e3T^{2}
43 1−0.780T+1.84e3T2 1 - 0.780T + 1.84e3T^{2}
47 1+58.1T+2.20e3T2 1 + 58.1T + 2.20e3T^{2}
53 1+6.29T+2.80e3T2 1 + 6.29T + 2.80e3T^{2}
59 1+79.3iT−3.48e3T2 1 + 79.3iT - 3.48e3T^{2}
61 1−97.5iT−3.72e3T2 1 - 97.5iT - 3.72e3T^{2}
67 1−61.4iT−4.48e3T2 1 - 61.4iT - 4.48e3T^{2}
71 1+72.5iT−5.04e3T2 1 + 72.5iT - 5.04e3T^{2}
73 1+65.6iT−5.32e3T2 1 + 65.6iT - 5.32e3T^{2}
79 1−152.T+6.24e3T2 1 - 152.T + 6.24e3T^{2}
83 1−13.4iT−6.88e3T2 1 - 13.4iT - 6.88e3T^{2}
89 1+44.0iT−7.92e3T2 1 + 44.0iT - 7.92e3T^{2}
97 1−26.8iT−9.40e3T2 1 - 26.8iT - 9.40e3T^{2}
show more
show less
   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

−11.31601055345491366661248824426, −10.19941966739331172497575355443, −9.458504133826006555674324002639, −7.964957712183780052117070154734, −7.17854560189193665204960194186, −6.57709244385942404552143443107, −4.96834720508582417032144504366, −4.53377319890528997869277400217, −3.13624471496315553461419209158, −0.828489051085108487486403810277, 0.26664419552179774332153009212, 2.37008638733329363992891206703, 4.05959608390791766789301137617, 4.89280187771612941276616378732, 6.05235907248709838862836570845, 6.71975528481630778607784228085, 8.010449685762454877295557830781, 8.722257407704124537193636467712, 9.992135859909670308924676476475, 10.85021329111716239842722280951

Graph of the ZZ-function along the critical line