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Properties

Label 2-105e2-1.1-c1-0-16
Degree 22
Conductor 1102511025
Sign 11
Analytic cond. 88.035088.0350
Root an. cond. 9.382709.38270
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  + 2·2-s + 2·4-s + 2·11-s − 13-s − 4·16-s + 19-s + 4·22-s − 2·26-s − 4·29-s + 9·31-s − 8·32-s − 3·37-s + 2·38-s + 10·41-s − 5·43-s + 4·44-s − 6·47-s − 2·52-s + 12·53-s − 8·58-s + 12·59-s + 10·61-s + 18·62-s − 8·64-s + 5·67-s + 6·71-s + 3·73-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.603·11-s − 0.277·13-s − 16-s + 0.229·19-s + 0.852·22-s − 0.392·26-s − 0.742·29-s + 1.61·31-s − 1.41·32-s − 0.493·37-s + 0.324·38-s + 1.56·41-s − 0.762·43-s + 0.603·44-s − 0.875·47-s − 0.277·52-s + 1.64·53-s − 1.05·58-s + 1.56·59-s + 1.28·61-s + 2.28·62-s − 64-s + 0.610·67-s + 0.712·71-s + 0.351·73-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 88.035088.0350
Root analytic conductor: 9.382709.38270
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 11025, ( :1/2), 1)(2,\ 11025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4970734164.497073416
L(12)L(\frac12) ≈\approx 4.4970734164.497073416
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)
bad3 1 1
5 1 1
7 1 1
good2 1−pT+pT2 1 - p T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−16.30327974501316, −15.77088208367699, −15.13756043914085, −14.71863529710234, −14.16354828067862, −13.69992059616406, −13.08309521323749, −12.62446257657459, −11.94074474769368, −11.56189889623774, −11.01258758001279, −10.03764871100801, −9.611615035496698, −8.771850003791022, −8.216967618378850, −7.230814823229727, −6.749655276700428, −6.084227448637340, −5.427385161911774, −4.854436244587081, −4.090734731637058, −3.618627563033512, −2.748416788514357, −2.056879787365213, −0.7832319802456344, 0.7832319802456344, 2.056879787365213, 2.748416788514357, 3.618627563033512, 4.090734731637058, 4.854436244587081, 5.427385161911774, 6.084227448637340, 6.749655276700428, 7.230814823229727, 8.216967618378850, 8.771850003791022, 9.611615035496698, 10.03764871100801, 11.01258758001279, 11.56189889623774, 11.94074474769368, 12.62446257657459, 13.08309521323749, 13.69992059616406, 14.16354828067862, 14.71863529710234, 15.13756043914085, 15.77088208367699, 16.30327974501316

Graph of the ZZ-function along the critical line