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Properties

Label 2-2960-1.1-c1-0-12
Degree 22
Conductor 29602960
Sign 11
Analytic cond. 23.635723.6357
Root an. cond. 4.861654.86165
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  − 3·3-s − 5-s + 3·7-s + 6·9-s − 5·11-s + 2·13-s + 3·15-s + 4·17-s + 4·19-s − 9·21-s − 6·23-s + 25-s − 9·27-s + 6·29-s + 4·31-s + 15·33-s − 3·35-s − 37-s − 6·39-s − 9·41-s − 10·43-s − 6·45-s + 11·47-s + 2·49-s − 12·51-s − 11·53-s + 5·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 1.13·7-s + 2·9-s − 1.50·11-s + 0.554·13-s + 0.774·15-s + 0.970·17-s + 0.917·19-s − 1.96·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.11·29-s + 0.718·31-s + 2.61·33-s − 0.507·35-s − 0.164·37-s − 0.960·39-s − 1.40·41-s − 1.52·43-s − 0.894·45-s + 1.60·47-s + 2/7·49-s − 1.68·51-s − 1.51·53-s + 0.674·55-s + ⋯

Functional equation

Λ(s)=(2960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29602960    =    24⋅5⋅372^{4} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 23.635723.6357
Root analytic conductor: 4.861654.86165
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2960, ( :1/2), 1)(2,\ 2960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88263639000.8826363900
L(12)L(\frac12) ≈\approx 0.88263639000.8826363900
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
37 1+T 1 + T
good3 1+pT+pT2 1 + p T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−11T+pT2 1 - 11 T + p T^{2}
53 1+11T+pT2 1 + 11 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−8.384495914592605943198193585522, −7.967716714125813453392078887841, −7.23620303213902323108245082851, −6.31278841946632243083099496990, −5.49557340632345683992432020156, −5.06756007617691541853072307264, −4.40343244869593798657772342377, −3.20508638243343033078558974345, −1.70436716945358454725317464833, −0.65053381069411396111812346191, 0.65053381069411396111812346191, 1.70436716945358454725317464833, 3.20508638243343033078558974345, 4.40343244869593798657772342377, 5.06756007617691541853072307264, 5.49557340632345683992432020156, 6.31278841946632243083099496990, 7.23620303213902323108245082851, 7.967716714125813453392078887841, 8.384495914592605943198193585522

Graph of the ZZ-function along the critical line