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Properties

Label 2-2664-2664.1627-c0-0-9
Degree 22
Conductor 26642664
Sign −0.241−0.970i-0.241 - 0.970i
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 + (0.309 − 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (−0.669 − 0.743i)6-s − 0.999·8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (−0.913 + 1.58i)11-s + (−0.978 + 0.207i)12-s + (0.669 + 1.15i)13-s + (−1.58 + 0.336i)15-s + (−0.5 + 0.866i)16-s + (−0.913 + 0.406i)18-s + (−0.809 + 1.40i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.309 − 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (−0.669 − 0.743i)6-s − 0.999·8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (−0.913 + 1.58i)11-s + (−0.978 + 0.207i)12-s + (0.669 + 1.15i)13-s + (−1.58 + 0.336i)15-s + (−0.5 + 0.866i)16-s + (−0.913 + 0.406i)18-s + (−0.809 + 1.40i)20-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((−0.241−0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((−0.241−0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: −0.241−0.970i-0.241 - 0.970i
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), −0.241−0.970i)(2,\ 2664,\ (\ :0),\ -0.241 - 0.970i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78880185770.7888018577
L(12)L(\frac12) ≈\approx 0.78880185770.7888018577
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+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
37 1+T 1 + T
good5 1+(0.809+1.40i)T+(−0.5+0.866i)T2 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.913−1.58i)T+(−0.5−0.866i)T2 1 + (0.913 - 1.58i)T + (-0.5 - 0.866i)T^{2}
13 1+(−0.669−1.15i)T+(−0.5+0.866i)T2 1 + (-0.669 - 1.15i)T + (-0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(0.978+1.69i)T+(−0.5+0.866i)T2 1 + (0.978 + 1.69i)T + (-0.5 + 0.866i)T^{2}
29 1+(−0.669+1.15i)T+(−0.5−0.866i)T2 1 + (-0.669 + 1.15i)T + (-0.5 - 0.866i)T^{2}
31 1+(0.809+1.40i)T+(−0.5+0.866i)T2 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2}
41 1+(0.309+0.535i)T+(−0.5+0.866i)T2 1 + (0.309 + 0.535i)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+(0.104−0.181i)T+(−0.5−0.866i)T2 1 + (0.104 - 0.181i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.104−0.181i)T+(−0.5+0.866i)T2 1 + (-0.104 - 0.181i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−1.82T+T2 1 - 1.82T + T^{2}
79 1+(−0.309+0.535i)T+(−0.5−0.866i)T2 1 + (-0.309 + 0.535i)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

−8.459016431585272066021486676050, −7.982972352271876918909242930813, −6.97740991569410954629234607208, −6.11791515852700168797391088651, −5.09381692415180741114686832077, −4.38782484501888655221490899415, −3.81785914986847178306566245125, −2.30143497232256746131695889117, −1.80507181127681388245733816366, −0.39576928961515880116085136204, 2.88816479287078064655101284575, 3.40973716538525059663136265987, 3.66022866521009936981407825682, 5.11517829568424293859302910867, 5.60955074570844839637840420935, 6.43899358511676640123495676495, 7.39720221201616405688051831075, 8.156127548937910287890925064483, 8.364967484440016063297594058068, 9.445710686037320084298626352794

Graph of the ZZ-function along the critical line