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Properties

Label 2-1083-1.1-c3-0-73
Degree 22
Conductor 10831083
Sign −1-1
Analytic cond. 63.899063.8990
Root an. cond. 7.993687.99368
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 7·4-s − 12·5-s − 3·6-s − 20·7-s − 15·8-s + 9·9-s − 12·10-s − 4·11-s + 21·12-s + 76·13-s − 20·14-s + 36·15-s + 41·16-s + 22·17-s + 9·18-s + 84·20-s + 60·21-s − 4·22-s + 82·23-s + 45·24-s + 19·25-s + 76·26-s − 27·27-s + 140·28-s − 242·29-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.577·3-s − 7/8·4-s − 1.07·5-s − 0.204·6-s − 1.07·7-s − 0.662·8-s + 1/3·9-s − 0.379·10-s − 0.109·11-s + 0.505·12-s + 1.62·13-s − 0.381·14-s + 0.619·15-s + 0.640·16-s + 0.313·17-s + 0.117·18-s + 0.939·20-s + 0.623·21-s − 0.0387·22-s + 0.743·23-s + 0.382·24-s + 0.151·25-s + 0.573·26-s − 0.192·27-s + 0.944·28-s − 1.54·29-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 63.899063.8990
Root analytic conductor: 7.993687.99368
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1083, ( :3/2), −1)(2,\ 1083,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
19 1 1
good2 1−T+p3T2 1 - T + p^{3} T^{2}
5 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
7 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
13 1−76T+p3T2 1 - 76 T + p^{3} T^{2}
17 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
23 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
29 1+242T+p3T2 1 + 242 T + p^{3} T^{2}
31 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
37 1−180T+p3T2 1 - 180 T + p^{3} T^{2}
41 1−390T+p3T2 1 - 390 T + p^{3} T^{2}
43 1−308T+p3T2 1 - 308 T + p^{3} T^{2}
47 1+522T+p3T2 1 + 522 T + p^{3} T^{2}
53 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
59 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
61 1+706T+p3T2 1 + 706 T + p^{3} T^{2}
67 1+104T+p3T2 1 + 104 T + p^{3} T^{2}
71 1−432T+p3T2 1 - 432 T + p^{3} T^{2}
73 1−718T+p3T2 1 - 718 T + p^{3} T^{2}
79 1+94T+p3T2 1 + 94 T + p^{3} T^{2}
83 1+1296T+p3T2 1 + 1296 T + p^{3} T^{2}
89 1+846T+p3T2 1 + 846 T + p^{3} T^{2}
97 1+830T+p3T2 1 + 830 T + p^{3} 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

−9.160271589402270976408109520017, −8.249962788334443005741485354870, −7.44081481822011889091134399463, −6.26919706364655544176094888769, −5.75525767772265521607407717880, −4.54961899759875406004252006029, −3.80244219032152589562823617417, −3.16252768986189697423571496283, −0.972523399241738064682902762509, 0, 0.972523399241738064682902762509, 3.16252768986189697423571496283, 3.80244219032152589562823617417, 4.54961899759875406004252006029, 5.75525767772265521607407717880, 6.26919706364655544176094888769, 7.44081481822011889091134399463, 8.249962788334443005741485354870, 9.160271589402270976408109520017

Graph of the ZZ-function along the critical line