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Properties

Label 2-2664-2664.1627-c0-0-0
Degree 22
Conductor 26642664
Sign 0.848−0.529i0.848 - 0.529i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
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  + (0.5 − 0.866i)2-s + (−0.809 + 0.587i)3-s + (−0.499 − 0.866i)4-s + (0.309 + 0.535i)5-s + (0.104 + 0.994i)6-s − 0.999·8-s + (0.309 − 0.951i)9-s + 0.618·10-s + (−0.669 + 1.15i)11-s + (0.913 + 0.406i)12-s + (−0.104 − 0.181i)13-s + (−0.564 − 0.251i)15-s + (−0.5 + 0.866i)16-s + (−0.669 − 0.743i)18-s + (0.309 − 0.535i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.809 + 0.587i)3-s + (−0.499 − 0.866i)4-s + (0.309 + 0.535i)5-s + (0.104 + 0.994i)6-s − 0.999·8-s + (0.309 − 0.951i)9-s + 0.618·10-s + (−0.669 + 1.15i)11-s + (0.913 + 0.406i)12-s + (−0.104 − 0.181i)13-s + (−0.564 − 0.251i)15-s + (−0.5 + 0.866i)16-s + (−0.669 − 0.743i)18-s + (0.309 − 0.535i)20-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.848−0.529i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((0.848−0.529i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.848−0.529i0.848 - 0.529i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(1627,⋅)\chi_{2664} (1627, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), 0.848−0.529i)(2,\ 2664,\ (\ :0),\ 0.848 - 0.529i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97487723510.9748772351
L(12)L(\frac12) ≈\approx 0.97487723510.9748772351
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
37 1+T 1 + T
good5 1+(−0.309−0.535i)T+(−0.5+0.866i)T2 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.669−1.15i)T+(−0.5−0.866i)T2 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.104+0.181i)T+(−0.5+0.866i)T2 1 + (0.104 + 0.181i)T + (-0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.913−1.58i)T+(−0.5+0.866i)T2 1 + (-0.913 - 1.58i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.104−0.181i)T+(−0.5−0.866i)T2 1 + (0.104 - 0.181i)T + (-0.5 - 0.866i)T^{2}
31 1+(−0.309−0.535i)T+(−0.5+0.866i)T2 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2}
41 1+(−0.809−1.40i)T+(−0.5+0.866i)T2 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.978−1.69i)T+(−0.5−0.866i)T2 1 + (0.978 - 1.69i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.978−1.69i)T+(−0.5+0.866i)T2 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−1.33T+T2 1 - 1.33T + T^{2}
79 1+(0.809−1.40i)T+(−0.5−0.866i)T2 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−9.582217395222222699501071346745, −8.644574690554241278461109553553, −7.32273988406750788877307680001, −6.65746958700625210386441604091, −5.70793966813640636419896794149, −5.10000914200769290649881939433, −4.45327425892102002172771013666, −3.44033045037640322299623649102, −2.61435072986306681337286900634, −1.37809701761524206132898157254, 0.62985556920797959521104501074, 2.31120160908014076597295096673, 3.43762923178478532971699423892, 4.72449770050354488101958759899, 5.12511763461191213133794369065, 5.96575781046311183078218785527, 6.47831838829440436704150021400, 7.33456324858770289695469786567, 8.066415582333957350191083536837, 8.711004557221679239983108117324

Graph of the ZZ-function along the critical line