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Properties

Label 2-644-644.111-c0-0-1
Degree 22
Conductor 644644
Sign −0.529+0.848i-0.529 + 0.848i
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  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (−0.841 + 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.959 − 0.281i)14-s + (−0.142 + 0.989i)16-s + (0.142 + 0.989i)18-s + (−1.25 − 1.45i)22-s + (0.841 + 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.654 + 0.755i)28-s + (0.544 + 0.627i)29-s + (0.841 + 0.540i)32-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (−0.841 + 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.959 − 0.281i)14-s + (−0.142 + 0.989i)16-s + (0.142 + 0.989i)18-s + (−1.25 − 1.45i)22-s + (0.841 + 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.654 + 0.755i)28-s + (0.544 + 0.627i)29-s + (0.841 + 0.540i)32-s + ⋯

Functional equation

Λ(s)=(644s/2 ΓC(s) L(s)=((−0.529+0.848i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(644s/2 ΓC(s) L(s)=((−0.529+0.848i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 644644    =    22⋅7⋅232^{2} \cdot 7 \cdot 23
Sign: −0.529+0.848i-0.529 + 0.848i
Analytic conductor: 0.3213970.321397
Root analytic conductor: 0.5669190.566919
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ644(111,⋅)\chi_{644} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 644, ( :0), −0.529+0.848i)(2,\ 644,\ (\ :0),\ -0.529 + 0.848i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.98500516120.9850051612
L(12)L(\frac12) ≈\approx 0.98500516120.9850051612
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+(−0.415+0.909i)T 1 + (-0.415 + 0.909i)T
7 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
23 1+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
good3 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
5 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
11 1+(−0.797+1.74i)T+(−0.654−0.755i)T2 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2}
13 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
17 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
19 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
29 1+(−0.544−0.627i)T+(−0.142+0.989i)T2 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2}
31 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
37 1+(−1.07−1.66i)T+(−0.415+0.909i)T2 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2}
41 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
43 1+(1.61−0.474i)T+(0.841−0.540i)T2 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−1.80+0.258i)T+(0.959−0.281i)T2 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2}
59 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
61 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
67 1+(−0.544−1.19i)T+(−0.654+0.755i)T2 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2}
71 1+(0.512−0.234i)T+(0.654−0.755i)T2 1 + (0.512 - 0.234i)T + (0.654 - 0.755i)T^{2}
73 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
79 1+(−0.118+0.822i)T+(−0.959−0.281i)T2 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2}
83 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
89 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
97 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)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

−10.65005074931358341455777410872, −9.891580955553663883073734294284, −8.768929181933916847063220236839, −8.198153462684166395386053044625, −6.66510200925045803854925561058, −5.83180212842871502472985087568, −4.77824658379035188934302503157, −3.64404626424773992238246294477, −2.88183659367968222508277016615, −1.08004331203658287825591957941, 2.42369779876652084127450154494, 3.71618036351885657688916811097, 4.81140634409601470531903045501, 5.72523856898641476924216991420, 6.57937541216715488396747831001, 7.35228117955439277874615288438, 8.495904321791663914227333439117, 9.191632722208906186421657152727, 9.776890057750255362141784989751, 11.38996316291292910691842521835

Graph of the ZZ-function along the critical line