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Properties

Label 2-160-40.13-c2-0-2
Degree 22
Conductor 160160
Sign −0.332−0.943i-0.332 - 0.943i
Analytic cond. 4.359684.35968
Root an. cond. 2.087982.08798
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  + (−0.130 + 0.130i)3-s + (−4.38 + 2.39i)5-s + (1.59 − 1.59i)7-s + 8.96i·9-s + 11.9i·11-s + (−9.59 + 9.59i)13-s + (0.260 − 0.887i)15-s + (0.857 − 0.857i)17-s − 20.5·19-s + 0.417i·21-s + (22.1 + 22.1i)23-s + (13.4 − 21.0i)25-s + (−2.34 − 2.34i)27-s + 27.3·29-s − 40.0·31-s + ⋯
L(s)  = 1  + (−0.0435 + 0.0435i)3-s + (−0.877 + 0.479i)5-s + (0.228 − 0.228i)7-s + 0.996i·9-s + 1.08i·11-s + (−0.738 + 0.738i)13-s + (0.0173 − 0.0591i)15-s + (0.0504 − 0.0504i)17-s − 1.08·19-s + 0.0198i·21-s + (0.963 + 0.963i)23-s + (0.539 − 0.841i)25-s + (−0.0870 − 0.0870i)27-s + 0.942·29-s − 1.29·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((−0.332−0.943i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+1) L(s)=((−0.332−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: −0.332−0.943i-0.332 - 0.943i
Analytic conductor: 4.359684.35968
Root analytic conductor: 2.087982.08798
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ160(113,⋅)\chi_{160} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :1), −0.332−0.943i)(2,\ 160,\ (\ :1),\ -0.332 - 0.943i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.539917+0.762793i0.539917 + 0.762793i
L(12)L(\frac12) ≈\approx 0.539917+0.762793i0.539917 + 0.762793i
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
5 1+(4.38−2.39i)T 1 + (4.38 - 2.39i)T
good3 1+(0.130−0.130i)T−9iT2 1 + (0.130 - 0.130i)T - 9iT^{2}
7 1+(−1.59+1.59i)T−49iT2 1 + (-1.59 + 1.59i)T - 49iT^{2}
11 1−11.9iT−121T2 1 - 11.9iT - 121T^{2}
13 1+(9.59−9.59i)T−169iT2 1 + (9.59 - 9.59i)T - 169iT^{2}
17 1+(−0.857+0.857i)T−289iT2 1 + (-0.857 + 0.857i)T - 289iT^{2}
19 1+20.5T+361T2 1 + 20.5T + 361T^{2}
23 1+(−22.1−22.1i)T+529iT2 1 + (-22.1 - 22.1i)T + 529iT^{2}
29 1−27.3T+841T2 1 - 27.3T + 841T^{2}
31 1+40.0T+961T2 1 + 40.0T + 961T^{2}
37 1+(−1.57−1.57i)T+1.36e3iT2 1 + (-1.57 - 1.57i)T + 1.36e3iT^{2}
41 1+37.5T+1.68e3T2 1 + 37.5T + 1.68e3T^{2}
43 1+(−49.2+49.2i)T−1.84e3iT2 1 + (-49.2 + 49.2i)T - 1.84e3iT^{2}
47 1+(−34.0+34.0i)T−2.20e3iT2 1 + (-34.0 + 34.0i)T - 2.20e3iT^{2}
53 1+(−28.8+28.8i)T−2.80e3iT2 1 + (-28.8 + 28.8i)T - 2.80e3iT^{2}
59 1−92.7T+3.48e3T2 1 - 92.7T + 3.48e3T^{2}
61 1−4.82iT−3.72e3T2 1 - 4.82iT - 3.72e3T^{2}
67 1+(54.6+54.6i)T+4.48e3iT2 1 + (54.6 + 54.6i)T + 4.48e3iT^{2}
71 1+59.2T+5.04e3T2 1 + 59.2T + 5.04e3T^{2}
73 1+(−34.1−34.1i)T+5.32e3iT2 1 + (-34.1 - 34.1i)T + 5.32e3iT^{2}
79 1−96.2iT−6.24e3T2 1 - 96.2iT - 6.24e3T^{2}
83 1+(63.6−63.6i)T−6.88e3iT2 1 + (63.6 - 63.6i)T - 6.88e3iT^{2}
89 1+3.68iT−7.92e3T2 1 + 3.68iT - 7.92e3T^{2}
97 1+(−46.0+46.0i)T−9.40e3iT2 1 + (-46.0 + 46.0i)T - 9.40e3iT^{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

−12.84369463834724353663668494558, −11.89374987722834512036115980976, −10.96064038006591122060749533554, −10.13980649073430894108779930377, −8.776297222349283889255150431266, −7.51411218384575551749050042423, −6.97414865053259331500352339959, −5.06579684524839429467756008804, −4.06906897173417865641900238928, −2.24651062898452295193772436094, 0.58762804741124329717946617472, 3.09579189414297609792862456427, 4.44729683334388917431073970581, 5.79528813756298342758545654047, 7.10767735189381235548712387980, 8.373061297091634734275981019421, 8.986974582799736243006804961035, 10.49105458622746309430492287240, 11.45794239954977175789751683833, 12.39057198813948275116837413627

Graph of the ZZ-function along the critical line