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Properties

Label 2-644-644.195-c0-0-1
Degree 22
Conductor 644644
Sign 0.603−0.797i0.603 - 0.797i
Analytic cond. 0.3213970.321397
Root an. cond. 0.5669190.566919
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (0.959 + 0.281i)9-s + (0.239 + 0.153i)11-s + (−0.142 − 0.989i)14-s + (−0.654 + 0.755i)16-s + (0.654 + 0.755i)18-s + (0.118 + 0.258i)22-s + (−0.959 + 0.281i)23-s + (−0.841 + 0.540i)25-s + (0.415 − 0.909i)28-s + (−0.698 − 1.53i)29-s + (−0.959 + 0.281i)32-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (0.959 + 0.281i)9-s + (0.239 + 0.153i)11-s + (−0.142 − 0.989i)14-s + (−0.654 + 0.755i)16-s + (0.654 + 0.755i)18-s + (0.118 + 0.258i)22-s + (−0.959 + 0.281i)23-s + (−0.841 + 0.540i)25-s + (0.415 − 0.909i)28-s + (−0.698 − 1.53i)29-s + (−0.959 + 0.281i)32-s + ⋯

Functional equation

Λ(s)=(644s/2 ΓC(s) L(s)=((0.603−0.797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(644s/2 ΓC(s) L(s)=((0.603−0.797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 644644    =    22⋅7⋅232^{2} \cdot 7 \cdot 23
Sign: 0.603−0.797i0.603 - 0.797i
Analytic conductor: 0.3213970.321397
Root analytic conductor: 0.5669190.566919
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ644(195,⋅)\chi_{644} (195, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 644, ( :0), 0.603−0.797i)(2,\ 644,\ (\ :0),\ 0.603 - 0.797i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4262126871.426212687
L(12)L(\frac12) ≈\approx 1.4262126871.426212687
L(1)L(1) 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+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
7 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
23 1+(0.959−0.281i)T 1 + (0.959 - 0.281i)T
good3 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
5 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
11 1+(−0.239−0.153i)T+(0.415+0.909i)T2 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2}
13 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
17 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
19 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
29 1+(0.698+1.53i)T+(−0.654+0.755i)T2 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2}
31 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
37 1+(−0.425+1.45i)T+(−0.841−0.540i)T2 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2}
41 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
43 1+(−0.273+1.89i)T+(−0.959−0.281i)T2 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.817−0.708i)T+(0.142−0.989i)T2 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2}
59 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
61 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
67 1+(0.698−0.449i)T+(0.415−0.909i)T2 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2}
71 1+(−1.07−1.66i)T+(−0.415+0.909i)T2 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2}
73 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
79 1+(−1.10+1.27i)T+(−0.142−0.989i)T2 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2}
83 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
89 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
97 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{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

−10.98159086231213536672194083891, −10.04966148117687477186386969782, −9.222557338681238592414041832958, −7.78503816095351405743086324151, −7.38028101660177667425654918050, −6.40855376916982194746710992602, −5.53159195079931390018441952677, −4.18210096799366597931473876415, −3.78017519468288730713362677628, −2.13638851526052235030581545632, 1.70321731499122355163687785107, 3.02464921740565158653968290842, 3.99550467749531639950049967792, 5.02818209323460077533481917708, 6.16439433008079971530569168618, 6.67661229051591541463817605022, 7.982255642059685181861509489900, 9.346896719593310270676999816702, 9.790095331077183246732334318377, 10.72869189528752176476262576347

Graph of the ZZ-function along the critical line