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Properties

Label 2-1520-1.1-c1-0-1
Degree 22
Conductor 15201520
Sign 11
Analytic cond. 12.137212.1372
Root an. cond. 3.483853.48385
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 5-s − 2.82·7-s − 0.999·9-s − 4.82·11-s + 0.585·13-s + 1.41·15-s − 0.828·17-s + 19-s + 4.00·21-s − 4·23-s + 25-s + 5.65·27-s + 4.82·29-s + 6.82·33-s + 2.82·35-s + 1.75·37-s − 0.828·39-s + 4.82·41-s + 2.82·43-s + 0.999·45-s − 8.48·47-s + 1.00·49-s + 1.17·51-s − 1.07·53-s + 4.82·55-s − 1.41·57-s + ⋯
L(s)  = 1  − 0.816·3-s − 0.447·5-s − 1.06·7-s − 0.333·9-s − 1.45·11-s + 0.162·13-s + 0.365·15-s − 0.200·17-s + 0.229·19-s + 0.872·21-s − 0.834·23-s + 0.200·25-s + 1.08·27-s + 0.896·29-s + 1.18·33-s + 0.478·35-s + 0.288·37-s − 0.132·39-s + 0.754·41-s + 0.431·43-s + 0.149·45-s − 1.23·47-s + 0.142·49-s + 0.164·51-s − 0.147·53-s + 0.651·55-s − 0.187·57-s + ⋯

Functional equation

Λ(s)=(1520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15201520    =    24⋅5⋅192^{4} \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 12.137212.1372
Root analytic conductor: 3.483853.48385
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1520, ( :1/2), 1)(2,\ 1520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.52345633130.5234563313
L(12)L(\frac12) ≈\approx 0.52345633130.5234563313
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
5 1+T 1 + T
19 1−T 1 - T
good3 1+1.41T+3T2 1 + 1.41T + 3T^{2}
7 1+2.82T+7T2 1 + 2.82T + 7T^{2}
11 1+4.82T+11T2 1 + 4.82T + 11T^{2}
13 1−0.585T+13T2 1 - 0.585T + 13T^{2}
17 1+0.828T+17T2 1 + 0.828T + 17T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−4.82T+29T2 1 - 4.82T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−1.75T+37T2 1 - 1.75T + 37T^{2}
41 1−4.82T+41T2 1 - 4.82T + 41T^{2}
43 1−2.82T+43T2 1 - 2.82T + 43T^{2}
47 1+8.48T+47T2 1 + 8.48T + 47T^{2}
53 1+1.07T+53T2 1 + 1.07T + 53T^{2}
59 1−2.82T+59T2 1 - 2.82T + 59T^{2}
61 1−9.65T+61T2 1 - 9.65T + 61T^{2}
67 1−6.58T+67T2 1 - 6.58T + 67T^{2}
71 1−7.31T+71T2 1 - 7.31T + 71T^{2}
73 1+16.1T+73T2 1 + 16.1T + 73T^{2}
79 1−14.8T+79T2 1 - 14.8T + 79T^{2}
83 1+8T+83T2 1 + 8T + 83T^{2}
89 1+8.82T+89T2 1 + 8.82T + 89T^{2}
97 1−6.24T+97T2 1 - 6.24T + 97T^{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

−9.678396174006840221426253358788, −8.546023845162878341187752307343, −7.901800735337600638582187594710, −6.90713023617083646657899514980, −6.14750277908761481683413296837, −5.44996691928555218222450854756, −4.55945215810712469391339379799, −3.36684813268620376838480250098, −2.51801468018039126786721732517, −0.50151682254725627847781573475, 0.50151682254725627847781573475, 2.51801468018039126786721732517, 3.36684813268620376838480250098, 4.55945215810712469391339379799, 5.44996691928555218222450854756, 6.14750277908761481683413296837, 6.90713023617083646657899514980, 7.901800735337600638582187594710, 8.546023845162878341187752307343, 9.678396174006840221426253358788

Graph of the ZZ-function along the critical line