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Properties

Label 2-2888-152.69-c0-0-1
Degree 22
Conductor 28882888
Sign 0.977+0.211i0.977 + 0.211i
Analytic cond. 1.441291.44129
Root an. cond. 1.200541.20054
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 + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 7-s + 0.999·8-s − 0.999·12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 7-s + 0.999·8-s − 0.999·12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s − 0.999·26-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((0.977+0.211i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((0.977+0.211i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: 0.977+0.211i0.977 + 0.211i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(69,⋅)\chi_{2888} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), 0.977+0.211i)(2,\ 2888,\ (\ :0),\ 0.977 + 0.211i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0334272681.033427268
L(12)L(\frac12) ≈\approx 1.0334272681.033427268
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
19 1 1
good3 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
7 1+T+T2 1 + T + T^{2}
11 1−T2 1 - T^{2}
13 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1−2T+T2 1 - 2T + T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
53 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−9.215519605372393620171009883480, −8.328063825257339153129734589739, −7.86032402923715331889017733720, −6.68499963886323168403976282472, −5.86465715922682318028569953637, −4.65619553197791220842898295159, −3.87600628666576127716305014746, −3.26799955148338571750394874975, −2.58768711863928477175259881496, −1.01751438959657666433319557663, 1.01175264422348499062057227176, 2.14781118742825836518892415704, 3.26816237801106861672720167164, 4.40054473952292998152017747488, 5.36197487237039507766942809976, 6.31225363905851397677899504655, 6.81415880743202642409991221328, 7.40840189571551137987850192177, 8.144504080311415666994128250952, 8.811986002893230565674364579946

Graph of the ZZ-function along the critical line