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Properties

Label 2-264-264.227-c0-0-1
Degree 22
Conductor 264264
Sign 0.970−0.242i0.970 - 0.242i
Analytic cond. 0.1317530.131753
Root an. cond. 0.3629780.362978
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  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + (0.809 − 0.587i)12-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s − 0.999·18-s + (−1.80 − 0.587i)19-s + (0.809 − 0.587i)22-s + 24-s + (−0.309 + 0.951i)25-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + (0.809 − 0.587i)12-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s − 0.999·18-s + (−1.80 − 0.587i)19-s + (0.809 − 0.587i)22-s + 24-s + (−0.309 + 0.951i)25-s + ⋯

Functional equation

Λ(s)=(264s/2 ΓC(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(264s/2 ΓC(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 264264    =    23⋅3⋅112^{3} \cdot 3 \cdot 11
Sign: 0.970−0.242i0.970 - 0.242i
Analytic conductor: 0.1317530.131753
Root analytic conductor: 0.3629780.362978
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ264(227,⋅)\chi_{264} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 264, ( :0), 0.970−0.242i)(2,\ 264,\ (\ :0),\ 0.970 - 0.242i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.98783362300.9878336230
L(12)L(\frac12) ≈\approx 0.98783362300.9878336230
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
good5 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
7 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
13 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
17 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
19 1+(1.80+0.587i)T+(0.809+0.587i)T2 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2}
23 1+T2 1 + T^{2}
29 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
41 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
43 1−1.17iT−T2 1 - 1.17iT - T^{2}
47 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
53 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
59 1+(−1.80+0.587i)T+(0.809−0.587i)T2 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1−0.618T+T2 1 - 0.618T + T^{2}
71 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
73 1+(−1.11+0.363i)T+(0.809−0.587i)T2 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
89 1−1.17iT−T2 1 - 1.17iT - T^{2}
97 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)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

−12.53754645894466774899537708340, −11.42029206930293426009731281449, −10.88320607462122253728744546217, −8.863944082603331619986402463646, −8.183784225832969237458163601321, −6.99820737920006169748928740343, −6.28867917791229953829435959466, −5.34806349899617237912977398325, −3.90665421499112695506344499848, −2.34333803514186724896722920634, 2.37362098857690372097412503603, 3.99531835212255375205442183375, 4.60400992516194197325476296894, 5.86363959105077971020893479102, 6.79997513346974898580798213337, 8.598551555791422644854415597380, 9.715693376172804525377898265638, 10.36469697255195961855380329691, 11.23219503082205516994490223784, 12.11760008892500796427656830928

Graph of the ZZ-function along the critical line