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Properties

Label 2-1127-1.1-c3-0-101
Degree 22
Conductor 11271127
Sign 11
Analytic cond. 66.495166.4951
Root an. cond. 8.154458.15445
Motivic weight 33
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 + 5·3-s − 4·4-s + 6·5-s − 10·6-s + 24·8-s − 2·9-s − 12·10-s + 34·11-s − 20·12-s + 57·13-s + 30·15-s − 16·16-s + 80·17-s + 4·18-s + 70·19-s − 24·20-s − 68·22-s + 23·23-s + 120·24-s − 89·25-s − 114·26-s − 145·27-s + 245·29-s − 60·30-s − 103·31-s − 160·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.962·3-s − 1/2·4-s + 0.536·5-s − 0.680·6-s + 1.06·8-s − 0.0740·9-s − 0.379·10-s + 0.931·11-s − 0.481·12-s + 1.21·13-s + 0.516·15-s − 1/4·16-s + 1.14·17-s + 0.0523·18-s + 0.845·19-s − 0.268·20-s − 0.658·22-s + 0.208·23-s + 1.02·24-s − 0.711·25-s − 0.859·26-s − 1.03·27-s + 1.56·29-s − 0.365·30-s − 0.596·31-s − 0.883·32-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: 11
Analytic conductor: 66.495166.4951
Root analytic conductor: 8.154458.15445
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1127, ( :3/2), 1)(2,\ 1127,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.3350572132.335057213
L(12)L(\frac12) ≈\approx 2.3350572132.335057213
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)
bad7 1 1
23 1−pT 1 - p T
good2 1+pT+p3T2 1 + p T + p^{3} T^{2}
3 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
5 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
13 1−57T+p3T2 1 - 57 T + p^{3} T^{2}
17 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
19 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
29 1−245T+p3T2 1 - 245 T + p^{3} T^{2}
31 1+103T+p3T2 1 + 103 T + p^{3} T^{2}
37 1+298T+p3T2 1 + 298 T + p^{3} T^{2}
41 1+95T+p3T2 1 + 95 T + p^{3} T^{2}
43 1−88T+p3T2 1 - 88 T + p^{3} T^{2}
47 1−357T+p3T2 1 - 357 T + p^{3} T^{2}
53 1+414T+p3T2 1 + 414 T + p^{3} T^{2}
59 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
61 1+822T+p3T2 1 + 822 T + p^{3} T^{2}
67 1−926T+p3T2 1 - 926 T + p^{3} T^{2}
71 1−335T+p3T2 1 - 335 T + p^{3} T^{2}
73 1−899T+p3T2 1 - 899 T + p^{3} T^{2}
79 1+1322T+p3T2 1 + 1322 T + p^{3} T^{2}
83 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
89 1−460T+p3T2 1 - 460 T + p^{3} T^{2}
97 1−964T+p3T2 1 - 964 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

−9.347265278831962154468950862376, −8.684281858709870916057383361474, −8.145623811305591597253945961623, −7.26630819644795689272943644300, −6.11369006916397024376182009991, −5.21386695427852631264990455798, −3.90220371078706883594373918953, −3.23881356246909613405600607272, −1.77857451426639356592294516615, −0.921343067813029429220734352762, 0.921343067813029429220734352762, 1.77857451426639356592294516615, 3.23881356246909613405600607272, 3.90220371078706883594373918953, 5.21386695427852631264990455798, 6.11369006916397024376182009991, 7.26630819644795689272943644300, 8.145623811305591597253945961623, 8.684281858709870916057383361474, 9.347265278831962154468950862376

Graph of the ZZ-function along the critical line