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Properties

Label 2-105e2-1.1-c1-0-18
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-s − 4-s − 3·8-s + 2·11-s + 6·13-s − 16-s + 6·17-s + 6·19-s + 2·22-s − 4·23-s + 6·26-s − 8·29-s + 6·31-s + 5·32-s + 6·34-s + 6·37-s + 6·38-s − 6·41-s − 2·44-s − 4·46-s − 6·52-s + 2·53-s − 8·58-s − 12·59-s + 6·62-s + 7·64-s − 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.603·11-s + 1.66·13-s − 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.426·22-s − 0.834·23-s + 1.17·26-s − 1.48·29-s + 1.07·31-s + 0.883·32-s + 1.02·34-s + 0.986·37-s + 0.973·38-s − 0.937·41-s − 0.301·44-s − 0.589·46-s − 0.832·52-s + 0.274·53-s − 1.05·58-s − 1.56·59-s + 0.762·62-s + 7/8·64-s − 0.488·67-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 3.0557756583.055775658
L(12)L(\frac12) ≈\approx 3.0557756583.055775658
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−T+pT2 1 - T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−14T+pT2 1 - 14 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−16.46013016401643, −15.87689909392640, −15.23063491818718, −14.69468231815389, −14.07692354919444, −13.60537236389792, −13.35373713450540, −12.43216857880638, −11.99213870197887, −11.52250586377623, −10.79139415353811, −9.944014044996831, −9.459889906081372, −8.949777106087611, −8.061708648427109, −7.775910130823162, −6.646171285232532, −6.020351097568707, −5.591082529048530, −4.886522192828421, −3.927730949920945, −3.600306650973034, −2.944056965893721, −1.559003025695040, −0.7959091287441254, 0.7959091287441254, 1.559003025695040, 2.944056965893721, 3.600306650973034, 3.927730949920945, 4.886522192828421, 5.591082529048530, 6.020351097568707, 6.646171285232532, 7.775910130823162, 8.061708648427109, 8.949777106087611, 9.459889906081372, 9.944014044996831, 10.79139415353811, 11.52250586377623, 11.99213870197887, 12.43216857880638, 13.35373713450540, 13.60537236389792, 14.07692354919444, 14.69468231815389, 15.23063491818718, 15.87689909392640, 16.46013016401643

Graph of the ZZ-function along the critical line