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Properties

Label 2-2e5-1.1-c1-0-0
Degree 22
Conductor 3232
Sign 11
Analytic cond. 0.2555210.255521
Root an. cond. 0.5054910.505491
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·9-s + 6·13-s + 2·17-s − 25-s − 10·29-s − 2·37-s + 10·41-s + 6·45-s − 7·49-s + 14·53-s − 10·61-s − 12·65-s − 6·73-s + 9·81-s − 4·85-s + 10·89-s + 18·97-s − 2·101-s + 6·109-s − 14·113-s − 18·117-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s + 1.66·13-s + 0.485·17-s − 1/5·25-s − 1.85·29-s − 0.328·37-s + 1.56·41-s + 0.894·45-s − 49-s + 1.92·53-s − 1.28·61-s − 1.48·65-s − 0.702·73-s + 81-s − 0.433·85-s + 1.05·89-s + 1.82·97-s − 0.199·101-s + 0.574·109-s − 1.31·113-s − 1.66·117-s + ⋯

Functional equation

Λ(s)=(32s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(32s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3232    =    252^{5}
Sign: 11
Analytic conductor: 0.2555210.255521
Root analytic conductor: 0.5054910.505491
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 32, ( :1/2), 1)(2,\ 32,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.65551438850.6555143885
L(12)L(\frac12) ≈\approx 0.65551438850.6555143885
L(32)L(\frac{3}{2}) 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 1
good3 1+pT2 1 + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−18T+pT2 1 - 18 T + p 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

−16.73856366990991880815681244917, −15.74882074786535250024495459074, −14.57652563978276046331230069557, −13.27687552535142704095990346642, −11.76661268274493420855693255102, −10.90769214371221130983350005998, −8.955386231165229198073332132052, −7.77199473906097062385997282225, −5.87146418848833687506982135026, −3.67478222653086463350186782835, 3.67478222653086463350186782835, 5.87146418848833687506982135026, 7.77199473906097062385997282225, 8.955386231165229198073332132052, 10.90769214371221130983350005998, 11.76661268274493420855693255102, 13.27687552535142704095990346642, 14.57652563978276046331230069557, 15.74882074786535250024495459074, 16.73856366990991880815681244917

Graph of the ZZ-function along the critical line